MECHANICAL METALLURGY
Metallurgy and Metallurgical Engineering Series
Robert F. Mehl, Consulting Editor
Michael B. Bever, Associate Consulting Editor
Barrett • Structure of Metals
BiRCHENALL • Physical Metallurgy
Bridgman • Studies in Large Plastic Flow and Fracture
Briggs • The Metallurgy of Steel Castings
Butts • Metallurgical Problems
Darken and Gurry • Physical Chemistry of Metals
Dieter • Mechanical Metallurgy
Gaudin • Flotation
Hansen • Constitution of Binary Alloys
Kehl • The Principles of Metallographic Laboratory Practice
Rhines • Phase Diagrams in Metallurgy
Seitz • The Physics of Metals
Smith • Properties of Metals at Elevated Temperatures
Williams and Homerberg • Principles of Metallography
VSL
Mechanical Metallurgy
GEORGE E. DIETER, JR.
Professor and Head of Department of Metallurgical
Engineering
Drexel Institute of Technologij
Philadelphia 4, Pa.
McGRAWHILL BOOK COMPANY
New York Toronto London 1961
MECHANICAL METALLURGY
Copyright © 1961 by the McGrawHill Book Company, Inc. Printed
in the United States of America. All rights reserved. This book, or
parts thereof, may not be reproduced in any form without permis
sion of the publishers. Library of Congress Catalog Number 6111385
12 13 14 15 16 MAMM  7 5 4 3
ISBN 070168903
PREFACE
Mechanical metallurgy is the area of knowledge which deals with the
behavior and response of metals to applied forces. Since it is not a pre
cisely defined area, it will mean different things to different persons. To
some it will mean mechanical properties of metals or mechanical testing,
others may consider the field restricted to the plastic working and shaping
of metals, while still others confine their interests to the more theoretical
aspects of the field, which merge with metal physics and physical metal
lurgy. Still another group may consider that mechanical metallurgy is
closely allied with applied mathematics and applied mechanics. In writ
ing this book an attempt has been made to cover, in some measure, this
great diversity of interests. The objective has been to include the entire
scope of mechanical metallurgy in one fairly comprehensive volume.
The book has been divided into four parts. Part One, Mechanical
Fundamentals, presents the mathematical framework for many of the
chapters which follow. The concepts of combined stress and strain are
reviewed and extended into three dimensions. Detailed consideration
of the theories of yielding and an introduction to the concepts of plas
ticity are given. No attempt is made to carry the topics in Part One to
the degree of completion required for original problem solving. Instead,
the purpose is to acquaint metallurgically trained persons with the mathe
matical language encountered in some areas of mechanical metallurgy.
Part Two, Metallurgical Fundamentals, deals with the structural aspects
of plastic deformation and fracture. Emphasis is on the atomistics of
flow and fracture and the way in which metallurgical structure affects
these processes. The concept of the dislocation is introduced early in
Part Two and is used throughout to provide qualitative explanations for
such phenomena as strain hardening, the yield point, dispersed phase
hardening, and fracture. A more mathematical treatment of the proper
ties of dislocations is given in a separate chapter. The topics covered in
Part Two stem from physical metallurgy. However, most topics are dis
cussed in greater detail and with a different emphasis than when they
are first covered in the usual undergraduate course in physical metal
lurgy. Certain topics that are more physical metallurgy than mechanical
VI rreracc
Prefe
metallurgy are included to provide continuity and the necessary back
ground for readers who have not studied modern physical metallurgy.
Part Three, Applications to Materials Testing, deals with the engineer
ing aspects of the common testing techniques of mechanical failure of
metals. Chapters are devoted to the tension, torsion, hardness, fatigue,
creep, and impact tests. Others take up the important subjects of
residual stresses and the statistical analysis of mechanicalproperty data.
In Part Three emphasis is placed on the interpretation of the tests and
on the effect of metallurgical variables on mechanical behavior rather
than on the procedures for conducting the tests. It is assumed that the
actual performance of these tests will be covered in a concurrent labora
tory course or in a separate course. Part Four, Plastic Forming of
Metals, deals with the common mechanical processes for producing use
ful metal shapes. Little emphasis is given to the descriptive aspects of
this subject, since this can best be covered by plant trips and illustrated
lectures. Instead, the main attention is given to the mechanical and
metallurgical factors which control each process such as forging, rolling,
extrusion, drawing, and sheetmetal forming.
This book is written for the senior or firstyear graduate student in
metallurgical or mechanical engineering, as well as for practicing engi
neers in industry. While most universities have instituted courses in
mechanical metallurgy or mechanical properties, there is a great diversity
in the material covered and in the background of the students taking
these courses. Thus, for the present there can be nothing like a stand
ardized textbook on mechanical metallurgy. It is hoped that the breadth
and scope of this book will provide material for these somewhat diverse
requirements. It is further hoped that the existence of a comprehensive
treatment of the field of mechanical metallurgy will stimulate the develop
ment of courses which cover the total subject.
Since this book is intended for college seniors, graduate students, and
practicing engineers, it is expected to become a part of their professional
library. Although there has been no attempt to make this book a hand
book, some thought has been given to providing abundant references to
the literature on mechanical metallurgy. Therefore, more references are
included than is usual in the ordinary textbook. References have been
given to point out derivations or analyses beyond the scope of the book,
to provide the key to further information on controversial or detailed
points, and to emphasize important papers which are worthy of further
study. In addition, a bibliography of general references will be found
at the end of each chapter. A collection of problems is included at the
end of the volume. This is primarily for the use of the reader who is
engaged in industry and who desires some check on his comprehension of
the material.
Preface vii
The task of writing this book has been mainly one of sifting and sorting
facts and information from the literature and the many excellent texts on
specialized aspects of this subject. To cover the breadth of material
found in this book would require parts of over 15 standard texts and
countless review articles and individual contributions. A conscientious
effort has been made throughout to give credit to original sources. For
the occasional oversights that may have developed during the "boiling
down process" the author offers his apologies. He is indebted to many
authors and publishers who consented to the reproduction of illustrations.
Credit is given in the captions of the illustrations.
Finally, the author wishes to acknowledge the many friends who
advised him in this work. Special mention should be given to Professor
A. W. Grosvenor, Drexel Institute of Technology, Dr. G. T. Home,
Carnegie Institute of Technology, Drs. T. C. Chilton, J. H. Faupel,
W. L. Phillips, W. I. Pollock, and J. T. Ransom of the du Pont Company,
and Dr. A. S. Nemy of the ThompsonRamoWooldridge Corp.
George E. Dieter, Jr.
CONTENTS
Preface v
List of Symbols xvii
Part One. Mechanical Fundamentals
1. Introduction 3
11. Scope of This Book 3
12. Strength of Materials — Basic Assumptions 5
13. Elastic and Plastic Behavior 6
14. Average Stress and Strain 7
15. Tensile Deformation of Ductile Metal 8
16. Ductile vs. Brittle Behavior 9
17. What Constitutes Failure? 10
18. Concept of Stress and the Types of Stress 13
19. Concept of Strain and the Types of Strain 15
2. Stress and Strain Relationships for Elastic Behavior 17
21. Introduction 1^
22. Description of Stress at a Point 17
23. State of Stress in Two Dimensions (Plane Stress) 19
24. Mohr's Circle of Stress— Two Dimensions 23
25. State of Stress in Three Dimensions 24
26. Mohr's Circle — Three Dimensions 27
27. Description of Strain at a Point 31
28. Measurement of Surface Strain 33
29. StressStrain Relations ... 35
210. Calculation of Stresses from Elastic Strains • • 39
211. Generalized StressStrain Relationships 41
212. Theory of Elasticity 43
213. Stress Concentration 46
214. Spherical and Deviator Components of Stress and Strain .... 50
215. Strain Energy ^2
3. Elements of the Theory of Plasticity 54
31. Introduction 54
32. The Flow Curve ^^
33. True Strain 57
ix
K Contents
34. Yielding Criteria for Ductile Metals 58
35. Combined Stress Tests 62
36. Octahedral Shear Stress and Shear Strain 65
37. Invariants of Stress and Strain 66
38. Basis of the Theories of Plasticity 67
39. Flow Theories 69
310. Deformation Theories 72
311. Twodimensional Plastic Flow — Plane Strain 73
312. Slipfield Theory 74
Part Two. Metallurgical Fundamentals
4. Plastic Deformation of Single Crystals 81
41. Introduction 81
42. Concepts of Crystal Geometry 82
43. Lattice Defects 85
44. Deformation by Slip 90
45. Slip in a Perfect Lattice 95
46. Slip by Dislocation Movement 97
47. Critical Resolved Shear Stress for Slip 99
48. Testing of Single Crystals 102
49. Deformation by Twinning 104
410. Stacking Faults 108
411. Deformation Bands and Kink Bands 110
412. Strain Hardening of Single Crystals Ill
5. Plastic Deformation of Polycrystalline Aggregates » . . . .118
51. Introduction 118
52. Grain Boundaries and Deformation 119
53. Lowangle Grain Boundaries 123
54. Solidsolution Hardening 128
55. Yieldpoint Phenomenon 132
56. Strain Aging 135
57. Strengthening from Secondphase Particles 137
58. Hardening Due to Point Defects 145
59. Strain Hardening and Cold Work 146
510. Bauschinger Effect 149
511. Preferred Orientation 150
512. Annealing of Coldworked Metal 153
513. Anneahng Textures 156
6. Dislocation Theory . . . o . 1 58
61. Introduction 158
62. Methods of Detecting Dislocations 158
63. Burgers Vector and the Dislocation Loop 162
64. Dislocations in the Facecentered Cubic Lattice 164
65. Dislocations in the Hexagonal Closepacked Lattice 169
66. Dislocations in the Bodycentered Cubic Lattice 169
67. Stress Field of a Dislocation 171
68. Forces on Dislocations 174
69. Forces between Dislocations 175
Contents xi
610. Dislocation Climb 177
611. Jogs in Dislocations 178
612. Dislocation and Vacancy Interaction 179
613. Dislocation — Foreignatom Interaction 181
614. Dislocation Sources 183
615. Multiplication of Dislocations — FrankRead Source 184
616. Dislocation Pileup 186
7. Fracture 190
71. Introduction 190
72. Types of Fracture in Metals 190
73. Theoretical Cohesive Strength of Metals 192
74. Griffith Theory of Brittle Fracture 194
75. Modifications of the Griffith Theory 197
76. Fracture of Single Crystals 199
77. Metallographic Aspects of Brittle Fracture 200
78. Dislocation Theories of Fracture 204
79. Delayed Yielding 209
710. Velocity of Crack Propagation ... 210
711. Ductile Fracture 211
712. Notch Effect in Fracture 213
713. Concept of the Fracture Curve 215
714. Classical Theory of the DuctiletoBrittle Transition 216
715. Fracture under Combined Stresses 218
716. Effect of High Hydrostatic Pressure on Fracture 219
8. Internal Friction 221
81. Introduction 221
82. Phenomenological Description of Internal Friction 222
83. Anelasticity . 224
84. Relaxation Spectrum 227
85. Grainboundary Relaxation 227
86. The Snoek Effect 229
87. Thermoelastic Internal Friction 229
88. Dislocation Damping 230
89. Damping Capacity 232
Part Three. Applications to Materials Testing
9. The Tension Test 237
91. Engineering StressStrain Curve 237
92. TruestressTruestrain Curve 243
93. Instability in Tension 248
94. Stress Distribution at the Neck 250
95. Strain Distribution in the Tensile Specimen 252
96. Effect of Strain Rate on Tensile Properties 254
97. Effect of Temperature on Tensile Properties 256
98. Combined Effect of Strain Rate and Temperature 258
99. Notch Tensile Test 260
910. Tensile Properties of Steels 262
911. Anisotropy of Tensile Properties 269
xii Contents
10. The Torsion Test 273
101. Introduction 273
102. Mechanical Properties in Torsion 273
103. Torsional Stresses for Large Plastic Strains 276
104. Types of Torsion Failures 278
105. Torsion Test vs. Tension Test 279
11. The Hardness Test 282
111. Introduction 282
112. Brinell Hardness 283
113. Meyer Hardness 284
114. Analysis of Indentation by a Spherical Indenter 286
115. Relationship between Hardness and the Tensileflow Curve .... 287
116. Vickers Hardness 289
117. Rockwell Hardness Test 290
118. Microhardness Tests 291
119. Hardnessconversion Relationships 292
1110. Hardness at Elevated Temperatures 293
12. Fatigue of Metals 296
121. Introduction 296
122. Stress Cycles 297
123. The SiV Curve 299
124. Statistical Nature of Fatigue 301
125. Structural Features of Fatigue 304
126. Theories of Fatigue 307
127. Efifect of Stress Concentration on Fatigue 310
128. Size Effect 314
129. Surface Effects and Fatigue 315
1210. Corrosion Fatigue 320
1211. Effect of Mean Stress on Fatigue 323
1212. Fatigue under Combined Stresses 326
1213. Overstressing and Understressing 327
1214. Effect of Metallurgical Variables on Fatigue Properties 329
1215. Effect of Temperature on Fatigue 332
13. Creep and Stress Rupture ° ° 335
131. The Hightemperature Materials Problem 335
132. The Creep Curve 336
133. The Stressrupture Test 341
134. Deformation at Elevated Temperature 342
135. Fracture at Elevated Temperature 345
136. Theories of Lowtemperature Creep 347
137. Theories of Hightemperature Creep 349
138. Presentation of Engineering Creep Data 354
139. Prediction of Longtime Properties 356
1310. Hightemperature Alloys 359
1311. Effect of Metallurgical Variables 363
1312. Creep under Combined Stresses 367
1313. Stress Relaxation 367
Contents xiii
14. Brittle Failure and Impact Testing 370
141. The Brittlefailure Problem 370
142. Notchedbar Impact Tests 371
143. Slowbend Tests 375
144. Specialized Tests for Transition Temperature 377
145. Significance of the Transition Temperature 379
146. Metallurgical Factors Affecting Transition Temperature 381
147. Effect of Section Size 384
148. Notch Toughness of Heattreated Steels 385
149. Temper Embrittlement 387
1410. Hydrogen Embrittlement 388
1411. Flow and Fracture under Very Rapid Rates of Loading 390
15. Residual Stresses 393
151. Origin of Residual Stresses 393
152. Effects of Residual Stresses 397
153. Mechanical Methods of Residualstress Measurement 398
154. Deflection Methods 403
155. Xray Determination of Residual Stress 407
156. Quenching Stresses 411
157. Surface Residual Stresses 415
158. Stress Relief 417
16. Statistics Applied to Materials Testing 419
161. Why Statistics? 419
162. Errors and Samples 420
163. Frequency Distribution 421
164. Measures of Central Tendency and Dispersion 424
165. The Normal Distribution 426
166. Extremevalue Distributions 430
167. Tests of Significance 432
168. Analysis of Variance 435
169. Statistical Design of Experiments 439
1610. Linear Regression 441
1611. Control Charts 442
1612. Statistical Aspects of Size Effect in Brittle Fracture 444
1613. Statistical Treatment of the Fatigue Limit 446
Part Four. Plastic Forming of Metals
17. General Fundamentals of Metalworking 453
171. Classification of Forming Processes 453
172. Effect of Temperature on Forming Processes 455
173. Effect of Speed of Deformation on Forming Processes 458
174. Effect of Metallurgical Structure on Forming Processes 459
175. Mechanics of Metal Forming 462
176. Work of Plastic Deformation 466
177. Formability Tests and Criteria 468
178. Friction in Forming Operations 470
179. Experimental Techniques for Forming Analysis 471
xlv Contents
18. Forging 473
181. Classification of Forging Processes 473
182. Forging Equipment 476
183. Deformation in Compression 479
184. Forging in Plane Strain with Coulomb Friction 481
185. Forging in Plane Strain with Sticking Friction 483
186. Forging of a Cylinder in Plane Strain 483
187. Forging Defects 484
188. Residual Stresses in Forgings 486
19. Rolling of Metals 488
191. Classification of Rolling Processes 488
192. RoUing Equipment 489
193. Hot RoUing 491
194. Cold Rolling 492
195. Rolling of Bars and Shapes 493
196. Forces and Geometrical Relationships in Rolling 494
197. Main Variables in RoUing 498
198. Deformation in Rolling 501
199. Defects in RoUed Products 502
1910. Residual Stresses in Rolled Products 503
1911. Theories of Cold Rolling 504
1912. Theories of Hot RoUing 508
1913. Torque and Horsepower .511
20. Extrusion 514
201. Classification of Extrusion Processes 514
202. Extrusion Equipment 517
203. Variables in Extrusion 518
204. Deformation in Extrusion . 522
205. Extrusion Defects 524
206. Extrusion under Ideal Conditions 525
207. Extrusion with Friction and Nonhomogeneous Deformation .... 526
208. Extrusion of Tubing 527
209. Production of Seamless Pipe and Tubing 529
21 . Rod, Wire, and Tube Drawing 532
211. Introduction 532
212. Rod and Wire Drawing 532
213. Defects in Rod and Wire 534
214. Variables in Wire Drawing 535
215. Wire Drawing without Friction 536
216. Wire Drawing with Friction 539
217. Tubedrawing Processes 541
218. Tube Sinking 542
219. Tube Drawing with a Stationary Mandrel 543
2110. Tube Drawing with a Moving Mandrel 545
2111. Residual Stresses in Rod, Wire, and Tubes 547
22. Sheetmetal Forming 549
221. Introduction 549
Contents xv
222. Forming Methods 550
223. Shearing and Blanking 555
224. Bending 557
225. Stretch Forming 562
226. Deep Drawing 563
227. Redrawing Operations 568
228. Ironing and Sinking 569
229. Defects in Formed Parts 57]
2210. Tests for Formability 57^
Appendix. Constants, and Conversion Factors 577
Problems , . 579
Answers to Selected Problems 599
Name Index 603
Subject Index 609
LIST OF SYMBOLS
A Area
a Linear distance
ao Interatomic spacing
B Constant
6 Width or breadth
b Burgers vector of a dislocation
C Generahzed constant
Cij Elastic coefficients
c Length of Griffith crack
D Diameter, grain diameter
E Modulus of elasticity for axial loading (Young's modulus)
e Conventional, or engineering, strain
exp Base of natural logarithms (= 2.718)
F Force per unit length on a dislocation line
/ Coefficient of friction
G Modulus of elasticity in shear (modulus of rigidity)
9 Crackextension force
H Activation energy
h Distance, usually in thickness direction
{h,k,l) Miller indices of a crystallographic plane
/ Moment of inertia
J Invariant of the stress deviator; polar moment of inertia
K Strength coefficient
Kf Fatiguenotch factor
Ki Theoretical stressconcentration factor
k Yield stress in pure shear
L Length
I, m, n Direction cosines of normal to a plane
In Natural logarithm
log Logarithm to base 10
Mb Bending moment
Mt Torsional moment, torque
xvili List of Symbols
m Strainrate sensitivity
N Number of cycles of stress or vibration
n Sti'ainhardening exponent
n' Generalized constant in exponential term
P Load or external force
p Pressure
q Reduction in area; plasticconstraint factor; notch sensitivity
index in fatigue
R Radius of curvature; stress ratio in fatigue; gas constant
r Radial distance
*Si Total stress on a plane before resolution into normal and shear
components
Sij Elastic compliance
s Standard deviation of a sample
T Temperature
Tm Melting point
t Time; thickness
tr Time for rupture
U Elastic strain energy
Uo Elastic strain energy per unit volume
u, V, IV Components of displacement in x, ij, and z directions
[uvw] Miller indices for a crystallographic direction
V Volume
V Velocity; coefficient of variation
W Work
Z ZenerHollomon parameter
a Linear coefficient of thermal expansion
a,0,d,(t) Generalized angles
r Line tension of a dislocation
7 Shear strain
A Volume strain or cubical dilatation; finite change
8 Deformation or elongation; deflection; logarithmic decrement
e Natural, or true, strain
e Significant, or effective, true strain
e Truestrain rate
im Minimum creep rate
7/ Efficiency; coefficient of viscosity
6 Dorn timetemperature parameter
K Bulk modulus or volumetric modulus of elasticity
A Interparticle spacing
X Lame's constant
ju Lode's stress parameter
List of Symbols xix
p Poisson's ratio; Lode's strain parameter
p Density
a Normal stress; the standard deviation of a population
(To Yield stress or yield strength
(Tq Yield stress in plane strain
CT Significant; or effective, true stress
01,02,03 Principal stresses
0' Stress deviator
0" Hydrostatic component of stress
(Ta Alternating, or variable, stress
o,„ Average principal stress; mean stress
ar Range of stress
a„ Ultimate tensile strength
ou, Working stress
T Shearing stress; relaxation time
<l> Airy stress function
i/' Specific damping capacity
Part One
MECHANICAL FUNDAMENTALS
Chapter 1
INTRODUCTION
11. Scope of This Book
Mechanical metallurgy is the area of metallurgy which is concerned
primarily with the response of metals to forces or loads. The forces
may arise from the use of the metal as a member or part in a structure
or machine, in which case it is necessary to know something about the
limiting values which can be withstood without failure. On the other
hand, the objective may be to convert a cast ingot into a more useful
shape, such as a flat plate, and here it is necessary to know the con
ditions of temperature and rate of loading which minimize the forces
that are needed to do the job.
Mechanical metallurgy is not a subject which can be neatly isolated
and studied by itself. It is a combination of many disciplines and many
approaches to the problem of understanding the response of materials to
forces. On the one hand is the approach used in reference to strength
of materials and in the theories of elasticity and plasticity, where a metal
is considered to be a homogeneous material whose mechanical behavior
can be rather precisely described on the basis of only a very few material
constants. This approach is the basis for the rational design of struc
tural members and machine parts, and the three topics of strength of
materials, elasticity, and plasticity are covered in Part One of this book
from a more generalized point of view than is usually considered in a first
course in strength of materials. The material covered in Chaps. 1 to 3
can be considered the mathematical framework on which much of the
remainder of the book rests. For students of engineering who have had
an advanced course in strength of materials or machine design, it proba
bly will be possible to skim rapidly over these chapters. However, for
most students of metallurgy and for practicing engineers in industry, it is
worth spending the time to become familiar with the mathematics pre
sented in Part One.
The theories of strength of materials, elasticity, and plasticity lose
much of their power when the structure of the metal becomes an impor
3
4 Mechanical Fundamentals [Chap. 1
tant consideration and it can no longer be considered as a homogeneous
medium. Examples of this are in the hightemperature behavior of
metals, where the metallurgical structure may continuously change with
time, or in the ductiletobrittle transition, which occurs in plain car
bon steel. The determination of the relationship between mechanical
behavior and structure (as detected chiefly with microscopic and Xray
techniques) is the main responsibility of the mechanical metallurgist.
When mechanical behavior is understood in terms of metallurgical struc
ture, it is generally possible to improve the mechanical properties or at
least to control them. Part Two of this book is concerned with the metal
lurgical fundamentals of the mechanical behavior of metals. Metallurgi
cal students will find that some of the material in Part Two has been
covered in a previous course in physical metallurgy, since mechanical
metallurgy is part of the broader field of physical metallurgy. However,
these subjects are considered in greater detail than is usually the case in
a first course in physical metallurgy. In addition, certain topics which
pertain more to physical metallurgy than mechanical metallurgy have
been included in order to provide continuity and to assist nonmetallurgi
cal students who may not have had a course in physical metallurgy.
The last three chapters of Part Two, especially Chap. 6, are concerned
primarily with atomistic concepts of the flow and fracture of metals.
Many of the developments in these areas have been the result of the
alliance of the solidstate physicist with the metallurgist. This is an
area where direct observation is practically impossible and definitive
experiments of an indirect nature are difficult to conceive. Moreover,
it is an area of intense activity in which the lifetime of a concept or theory
may be rather short. Therefore, in writing these chapters an attempt
has been made to include only material which is generally valid and to
minimize the controversial aspects of the subject.
Basic data concerning the strength of metals and measurements for the
routine control of mechanical properties are obtained from a relatively
small number of standardized mechanical tests. Part Three, Appli
cations to Materials Testing, considers each of the common mechanical
tests, not from the usual standpoint of testing techniques, but instead
from the consideration of what these tests tell about the service per
formance of metals and how metallurgical variables affect the results of
these tests. Much of the material in Parts One and Two has been uti
lized in Part Three. It is assumed that the reader either has completed
a conventional course in materials testing or will be concurrently taking
a laboratory course in which familiarization with the testing techniques
will be acquired.
Part Four considers the metallurgical and mechanical factors involved
in the forming of metals into useful shapes. Attempts have been made
Sec. 12] Introduction 5
to present mathematical analyses of the principal metalworking processes,
although in certain cases this has not been possible, either because of the
considerable detail required or because the analysis is beyond the scope
of this book. No attempt has been made to include the extensive special
ized technology associated with each metalworking process, such as roll
ing or extrusion, although some effort has been made to give a general
impression of the mechanical equipment required and to familiarize the
reader with the specialized vocabulary of the metalworking field. Major
emphasis has been placed on presenting a fairly simplified picture of the
forces involved in each process and how geometrical and metallurgical
factors affect the forming loads and the success of the metalworking
process.
12. Strength of Materials — Basic Assumptions
Strength of materials is the body of knowledge which deals with the
relation betw^een internal forces, deformation, and external loads. In
the general method of analysis used in strength of materials the first step
is to assume that the member is in equilibrium. The equations of static
equilibrium are applied to the forces acting on some part of the body in
order to obtain a relationship between the external forces acting on the
member and the internal forces resisting the action of the external loads.
Since the equations of equilibrium must be expressed in terms of forces
acting external to the body, it is necessary to make the internal resisting
forces into external forces. This is done by passing a plane through the
body at the point of interest. The part of the body lying on one side
of the cutting plane is removed and replaced by the forces it exerted on
the cut section of the part of the body that remains. Since the forces
acting on the "free body" hold it in equilibrium, the equations of equi
librium may be applied to the problem.
The internal resisting forces are usually expressed by the stress^ acting
over a certain area, so that the internal force is the integral of the stress
times the differential area over which it acts. In order to evaluate this
integral, it is necessary to know the distribution of the stress over the
area of the cutting plane. The stress distribution is arrived at by observ
ing and measuring the strain distribution in the member, since stress
cannot be physically measured. However, since stress is proportional to
strain for the small deformations involved in most work, the determi
nation of the strain distribution provides the stress distribution. The
expression for the stress is then substituted into the equations of equi
* For present purposes stress is defined as force per unit area. The companion term
strain is defined as the change in length per unit length. More complete definitions
will be given later.
6 Mechanical Fundamentals [Chap. 1
librium, and they are solved for stress in terms of the loads and dimen
sions of the member.
Important assumptions in strength of materials are that the body
which is being analyzed is continuous, homogeneous, and isotropic. A
continuous body is one which does not contain voids or empty spaces of
any kind. A body is homogeneous if it has identical properties at all
points. A body is considered to be isotropic with respect to some property
when that property does not vary with direction or orientation. A
property which varies with orientation with respect to some system of
axes is said to be anisotropic.
While engineering materials such as steel, cast iron, and aluminum may
appear to meet these conditions when viewed on a gross scale, it is readily
apparent when they are viewed through a microscope that they are any
thing but homogeneous and isotropic. Most engineering metals are made
up of more than one phase, with different mechanical properties, such
that on a micro scale they are heterogeneous. Further, even a single
phase metal will usually exhibit chemical segregation, and therefore the
properties will not be identical from point to point. Metals are made up
of an aggregate of crystal grains having different properties in different
crystallographic directions. The reason why the equations of strength
of materials describe the behavior of real metals is that, in general, the
crystal grains are so small that, for a specimen of any macroscopic vol
ume, the materials are statistically homogeneous and isotropic. How
ever, when metals are severely deformed in a particular direction, as in
rolling or forging, the mechanical properties may be anisotropic on a
macro scale.
13. Elastic and Plastic Behavior
Experience shows that all solid materials can be deformed when sub
jected to external load. It is further found that up to certain limiting
loads a solid will recover its original dimensions when the load is removed.
The recovery of the original dimensions of a deformed body when the
load is removed is known as elastic behavior. The limiting load beyond
which the material no longer behaves elastically is the elastic limit. If
the elastic limit is exceeded, the body will experience a permanent set or
deformation when the load is removed. A body which is permanently
deformed is said to have undergone plastic deformation.
For most materials, as long as the load does not exceed the elastic
limit, the deformation is proportional to the load. This relationship is
known as Hooke's law; it is more frequently stated as stress is proportional
to strain. Hooke's law requires that the loaddeformation relationship
should be linear. However, it does not necessarily follow that all mate
Sec. 14]
Introduction
rials which behave elastically will have a linear stressstrain relationship.
Rubber is an example of a material with a nonlinear stressstrain relation
ship that still satisfies the definition of an elastic material.
Elastic deformations in metals are quite small and require very sensi
tive instruments for their measurement. Ultrasensitive instruments
have shown that the elastic limits of metals are much lower than the
values usually measured in engineering tests of materials. As the meas
uring devices become more sensitive, the elastic limit is decreased, so that
for most metals there is only a rather narrow range of loads over which
Hooke's law strictly applies. This is, however, primarily of academic
importance. Hooke's law remains a quite valid relationship for engi
neering design.
•(_4. Average Stress and Strain
As a starting point in the discussion of stress and strain, consider a
uniform cylindrical bar which is subjected to an axial tensile load (Fig.
11). Assume that two gage marks are put on the surface of the bar in
Lo + S
 Ln
^P
'a dA
^P
Fig. 11. Cylindrical bar subjected to Fig. 12. Freebody diagram for Fig. 11.
axial load.
its unstrained state and that Lo is the gage length between these marks.
A load P is applied to one end of the bar, and the gage length undergoes
a slight increase in length and decrease in diameter. The distance
between the gage marks has increased by an amount 5, called the defor
mation. The average linear strain e is the ratio of the change in length
to the original length.
_ J^ _ AL _ L  Lo
Lo Lo Lo
(11)
Strain is a dimensionless quantity since both 5 and Lo are expressed in
units of length.
Figure 12 shows the freebody diagram for the cylindrical bar shown
in Fig. 11. The external load P is balanced by the internal resisting
force j(7 dA, where a is the stress normal to the cutting plane and A is
8 Mechanical Fundamentals [Chap. 1
the crosssectional area of the bar. The equinbrium equation is
P = jadA (12)
If the stress is distributed uniformly over the area A, that is, if a is con
stant, Eq. (12) becomes
P = aj dA = (tA
cr = j (13)
In general, the stress will not be uniform over the area A, and therefore
Eq. (13) represents an average stress. For the stress to be absolutely
uniform, every longitudinal element in the bar would have to experience
exactly the same strain, and the proportionality between stress and strain
would have to be identical for each element. The inherent anisotropy
between grains in a polycrystalline metal rules out the possibility of com
plete uniformity of stress over a body of macroscopic size. The presence
of more than one phase also gives rise to nonuniformity of stress on a
microscopic scale. If the bar is not straight or not centrally loaded, the
strains will be different for certain longitudinal elements and the stress
will not be uniform. An extreme disruption in the uniformity of the
stress pattern occurs when there is an abrupt change in cross section.
This results in a stress raiser or stress concentration (see Sec. 213).
In engineering practice, the load is usually measured in pounds and
the area in square inches, so that stress has units of pounds per square
inch (psi). Since it is common for engineers to deal with loads in the
thousands of pounds, an obvious simpHfication is to work with units of
1,000 lb, called kips. The stress may be expressed in units of kips per
square inch (ksi). (1 ksi = 1,000 psi.) In scientific work stresses are
often expressed in units of kilograms per square millimeter or dynes per
square centimeter. (1 kg/mm^ = 9.81 X 10^ dynes/cml)
Below the elastic limit Hooke's law can be considered vaHd, so that
the average stress is proportional to the average strain,
 = E = constant (14)
e
The constant E is the modulus of elasticity, or Young's modulus.
15. Tensile Deformation of Ductile Metal
The basic data on the mechanical properties of a ductile metal are
obtained from a tension test, in which a suitably designed specimen is
subjected to increasing axial load until it fractures. The load and elonga
tion are measured at frequent intervals during the test and are expressed
Sec. 16]
Introduction
Fig. 13.
curve.
Fracture
Strain e
Typical tension stressstrain
as average stress and strain according to the equations in the previous
section. (More complete details on the tension test are given in Chap. 9.)
The data obtained from the tension test are generally plotted as a
stressstrain diagram. Figure 13 shows a typical stressstrain curve for
a metal such as aluminum or cop
per. The initial linear portion of
the curve OA is the elastic region
within which Hooke's law is
obeyed. Point A is the elastic
limit, defined as the greatest stress
that the metal can withstand with
out experiencing a permanent
strain when the load is removed.
The determination of the elastic
limit is quite tedious, not at all
routine, and dependent on the sen
sitivity of the strainmeasuring
instrument. For these reasons it is often replaced by the proportional
limit, point A'. The proportional limit is the stress at which the stress
strain curve deviates from linearity. The slope of the stressstrain curve
in this region is the modulus of elasticity.
For engineering purposes the limit of usable elastic behavior is described
by the yield strength, point B. The yield strength is defined as the stress
which will produce a small amount of permanent deformation, generally
a strain equal to 0.2 per cent or 0.002 inches per inch. In Fig. 13
this permanent strain, or offset, is OC. Plastic deformation begins
when the elastic limit is exceeded. As the plastic deformation of the
specimen increases, the metal becomes stronger (strain hardening) so
that the load required to extend the specimen increases with further
straining. Eventually the load reaches a maximum value. The maxi
mum load divided by the original area of the specimen is the ultimate
tensile strength. For a ductile metal the diameter of the specimen
begins to decrease rapidly beyona maximum load, so that the load
required to continue deformation drops off until the specimen fractures.
Since the average stress is based on the original area of the specimen,
it also decreases from maximum load to fracture.
16. Ductile vs. Brittle Behavior
The general behavior of materials under load can be classified as ductile
or brittle depending upon whether or not the material exhibits the ability
to undergo plastic deformation. Figure 13 illustrates the tension stress
strain curve of a ductile material. A completely brittle material would
10
Mechanical Fundamentals
[Chap. 1
fracture almost at the elastic limit (Fig. 14.0), while a brittle metal, such
as white cast iron, shows some slight measure of plasticity before fracture
(Fig. 146). Adequate ductility is an important engineering consider
ation, because it allows the mate
rial to redistribute localized stresses.
When localized stresses at notches
and other accidental stress concen
trations do not have to be con
sidered, it is possible to design for
static situations on the basis of
average stresses. However, with
brittle materials, localized stresses
continue to build up when there
is no local yielding. Finally, a
crack forms at one or more points
of stress concentration, and it
spreads rapidly over the section. Even if no stress concentrations are
present in a brittle material, fracture will still occur suddenly because the
yield stress and tensile strength are practically identical.
It is important to note that brittleness is not an absolute property of
a metal. A metal such as tungsten, which is brittle at room temperature,
is ductile at an elevated temperature. A metal which is brittle in tension
may be ductile under hydrostatic compression. Furthermore, a metal
which is ductile in tension at room temperature can become brittle in the
presence of notches, low temperature, high rates of loading, or embrittling
agents such as hydrogen.
strain
Fig. 14. (a) Stressstrain curve for com
pletely brittle material (ideal behavior);
(b) stressstrain curve for brittle metal
with slight amount of ductility.
17. What Constitutes Failure?
Structural members and machine elements can fail to perform their
intended functions in three general ways:
1. Excessive elastic deformation
2. Yielding, or excessive plastic deformation
3. Fracture
An understanding of the common types of failure is important in good
design because it is always necessary to relate the loads and dimensions
of the member to some significant material parameter which limits the
loadcarrying capacity of the member. For different types of failure,
different significant parameters will be important.
Two general types of excessive elastic deformation may occur: (1)
excessive deflection under conditions of stable equilibrium, such as the
Sec. 17] Introduction 11
deflection of beam under gradually applied loads; (2) sudden deflection,
or buckling, under conditions of unstable equilibrium.
Excessive elastic deformation of a machine part can mean failure of the
machine just as much as if the part completely fractured. For example,
a shaft which is too flexible can cause rapid wear of the bearing, or the
excessive deflection of closely mating parts can result in interference and
damage to the parts. The sudden buckling type of failure may occur in
a slender column when the axial load exceeds the Euler critical load or
when the external pres.sure acting against a thinwalled shell exceeds a
critical value. Failures due to excessive elastic deformation are con
trolled by the modulus of elasticity, not by the strength of the material.
Generally, little metallurgical control can be exercised over the elastic
modulus. The most effective way to increase the stiffness of a member
is usually by changing its shape and increasing the dimensions of its
cross section.
Yielding, or excessive plastic deformation, occurs when the elastic limit
of the metal has been exceeded. Yielding produces permanent change of
shape, which may prevent the part from functioning properly any longer.
In a ductile metal under conditions of static loading at room temperature
yielding rarely results in fracture, because the metal strain hardens as it
deforms, and an increased stress is required to produce further deforma
tion. Failure by excessive plastic deformation is controlled by the yield
strength of the metal for a uniaxial condition of loading. For more com
plex loading conditions the yield strength is still the significant parameter,
but it must be used with a suitable failure criterion (Sec. 34). At tem
peratures significantly greater than room temperature metals no longer
exhibit strain hardening. Instead, metals can continuously deform at
constant stress in a timedependent yielding known as creep. The failure
criterion under creep conditions is complicated by the fact that stress is
not proportional to strain and the further fact that the mechanical proper
ties of the material may change appreciably during service. This com
plex phenomenon will be considered in greater detail in Chap. 13.
The formation of a crack which can result in complete disruption of
continuity of the member constitutes fracture. A part made from a
ductile metal which is loaded statically rarely fractures like a tensile
specimen, because it will first fail by excessive plastic deformation.
However, metals fail by fracture in three general ways: (1) sudden
brittle fracture; (2) fatigue, or progressive fracture; (3) delayed fracture.
In the previous section it was shown that a brittle material fractures
under static loads with little outward evidence of yielding. A sudden
brittle type of fracture can also occur in ordinarily ductile metals under
certain conditions. Plain carbon structural steel is the most common
12 Mechanical Fundamentals [Chap. 1
example of a material with a ductiletobrittle transition. A change from
the ductile to the brittle type of fracture is promoted by a decrease in
temperature, an increase in the rate of loading, and the presence of a
complex state of stress due to a notch. This problem is considered in
Chap. 14.
Most fractures in machine parts are due to fatigue. Fatigue failures
occur in parts which are subjected to alternating, or fluctuating, stresses.
A minute crack starts at a localized spot, generally at a notch or stress
concentration, and gradually spreads over the cross section until the
member breaks. Fatigue failure occurs without any visible sign of yield
ing at nominal or average stresses that are well below the tensile strength
of the metal. Fatigue failure is caused by a critical localized tensile stress
which is very difficult to evaluate, and therefore design for fatigue failure
is based primarily on empirical relationships using nominal stresses.
Fatigue of metals is discussed in greater detail in Chap. 12.
One common type of delayed fracture is stressrupture failure, which
occurs when a metal has been statically loaded at an elevated temper
ature for a long period of time. Depending upon the stress and the tem
perature there may be no yielding prior to fracture. A similar type of
delayed fracture, in which there is no warning by yielding prior to failure,
occurs at room temperature when steel is statically loaded in the presence
of hydrogen.
All engineering materials show a certain variability in mechanical
properties, which in turn can be influenced by changes in heat treat
ment or fabrication. Further, uncertainties usually exist regarding the
magnitude of the applied loads, and approximations are usually neces
sary in calculating the stresses for all but the most simple member.
Allowance must be made for the possibility of accidental loads of high
magnitude. Thus, in order to provide a margin of safety and to protect
against failure from unpredictable causes, it is necessary that the allow
able stresses be smaller than the stresses which produce failure. The
value of stress for a particular material used in a particular way which is
considered to be a safe stress is usually called the working stress (Tw. For
static applications the working stress of ductile metals is usually based
on the yield strength ao and for brittle metals on the ultimate tensile
strength o,,. Values of working stress are established by local and Federal
agencies and by technical organizations such as the American Society of
Mechanical Engineers (ASME). The working stress may be considered
as either the yield strength or the tensile strength divided by a number
called the factor of safety.
Ow "^ jTf or CTu, = rj (l5j
•/V iV u
Sec. 18]
Introduction
13
where ay, = working stress, psi
cro = yield strength, psi
(r„ = tensile strength, psi
iVo = factor of safety based on yield strength
Nu = factor of safety based on tensile strength
The value assigned to the factor of safety depends on an estimate of
all the factors discussed above. In addition, careful consideration should
be given to the consequences which would result from failure. If failure
would result in loss of life, the factor of safety should be increased. The
type of equipment will also influence the factor of safety. In military
equipment, where light weight may be a prime consideration, the factor
of safety may be lower than in commercial equipment. The factor of
safety will also depend on the ex
pected type of loading. For static
loading, as in a building, the factor
of safety would be lower than in a
machine, which is subjected to vi p
bration and fluctuating stresses.
A
® L
u
\
f
k"^ 5
^\
\
~
V.
\
^o>
\^
•^
X^X
18. Concept of Stress a
Types of Stresses
nd tfi(
®
y
Pa
©
/'''^
\
\
\^
\
A — >x
Stress is defined as the internal
resistance of a body to an external
applied force, per unit area. In Sec.
14 the stress was considered to be
uniformly distributed over the cross
sectional area of the member. How
ever, this is not the general case.
Figure l5a represents a body in
equilibrium under the action of
external forces Pi, P2, . . . , Ph
There are two kinds of external
forces which may act on a body, sur
face forces and body forces. Forces
distributed over the surface of the
body, such as hydrostatic pressure
or the pressure exerted by one body
on another, are called surface forces.
Forces distributed over the volume
of a body, such as gravitational forces, magnetic forces, or inertia forces
(for a body in motion), are called body forces. The two most common
types of body forces encountered in engineering practice are centrifugal
Fig. 15. (a) Body in equilibrium under
action of external forces Pi, . . . , Pb',
(b) forces acting on parts.
14; Mechanical Fundamentals [Chap, 1
forces due to highspeed rotation and forces due to temperature diffe)'
ential over the body (thermal stress).
In general the force will not be uniformly distributed over any cross
section of the body illustrated in Fig. l5a. To obtain the stress at some
point in a plane such as mm, part 1 of the body is removed and replaced
by the system of external forces on mm which will retain each point in
part 2 of the body in the same position as before the removal of part 1.
Fig. 16. Resolution of total stress into its components.
This is the situation in Fig. 156. We then take an area AA surrounding
the point and note that a force AP acts on this area. If the area AA is
continuously reduced to zero, the limiting value of the ratio AP/AA is the
stress at the point on plane mm of body 2.
lim — r = cr (16)
AA^O AA
The stress will be in the direction of the resultant force P and will gener
ally be inclined at an angle to A A. The same stress at point in plane
mm would be obtained if the free body were constructed by removing
part 2 of the solid body. However, the stress will be different on any
other plane passing through point 0, such as the plane nn.
It is inconvenient to use a stress which is inclined at some arbitrary
angle to the area over which it acts. The total stress can be resolved
into two components, a normal stress a perpendicular to AA, and a shear
ing stress (or shear stress) r lying in the plane mm of the area. To illus
trate this point, consider Fig. 16. The force P makes an angle 6 with
the normal z to the plane of the area A. Also, the plane containing the
normal and P intersects the plane A along a dashed line that makes an
angle <^ with the y axis. The normal stress is given by
P
a ^ r COS 6 (17)
A
Sec. 19] Introduction 15
The shear stress in the plane acts along the line OC and has the magnitude
T=T&\nd (18)
This shear stress may be further resolved into components parallel to
the X and y directions lying in the plane.
P
X direction "^ ~ ~a ^^^ ^ ^^^^ ^ (1"9)
P
y direction '^ ~ ~a ^^^^ ^ ^^^ ^ (110)
Therefore, in general a given plane may have one normal stress and two
shear stresses acting on it.
19. Concept of Strain and the Types of Strain
In Sec. 14 the average linear strain was defined as the ratio of the
change in length to the original length of the same dimension.
_ ± _ AL L  Ln
where e = average linear strain
5 = deformation
By analogy with the definition of stress at a point, the strain at a point
is the ratio of the deformation to the gage length as the gage length
approaches zero.
Rather than referring the change in length to the original gage length,
it often is more useful to define the strain as the change in linear dimen
sion divided by the instantaneous value of the dimension.
— = \nj^ (111)
The above equation defines the natural, or true, strain. True strain,
which is useful in dealing with problems in plasticity and metal form
ing, will be discussed more fully in Chap. 3. For the present it should
be noted that for the very small strains for which the equations of elas
ticity are valid the two definitions of strain give identical values.^
Not only will the elastic deformation of a body result in a change in
length of a linear element in the body, but it may also result in a change
^ Considerable variance exists in the literature regarding the notation for average
linear strain, true strain, and deformation. Linear strain is often denoted by e, while
true strain is sometimes denoted by 5 or e.
16
Mechanical Fundamentals
[Chap. 1
in the initial angle between any two lines. The angular change in a
right angle is known as shear strain. Figure 17 illustrates the strain
produced by the pure shear of one face of a cube. The angle at A, which
was originally 90°, is decreased by the application of a shear stress by a
small amount 6. The shear strain y is equal to the displacement a
Fig. 17. Shear strain.
divided by the distance between the planes, h. The ratio a/h is also
the tangent of the angle through which the element has been rotated.
For the small angles usually involved, the tangent of the angle and the
angle (in radians) are equal. Therefore, shear strains are often expressed
as angles of rotation.
7 = 7 = tan 6 =
n
(112)
BIBLIOGRAPHY
Crandall, S. H., and N. C. Dahl (eds.): "An Introduction to the Mechanics of Solids,"
McGrawHill Book Company, Inc., New York, 1959.
Frocht, M. M.: "Strength of Materials," The Ronald Press Company, New York,
1951.
Seely, F. B., and J. O. Smith: "Resistance of Materials," 4th ed., John Wiley & Sons,
Inc., New York, 1957.
and ^: "Advanced Mechanics of Materials," 2d ed., John Wiley &
Sons, Inc., New York, 1952.
Shanley, F. R.: "Strength of Materials," McGrawHill Book Company, Inc., New
York, 1957.
Chapter 2
STRESS AND STRAIN RELATIONSHIPS
FOR ELASTIC BEHAVIOR
21 . Introduction
The purpose of this chapter is to present the mathematical relation
ships for expressing the stress and strain at a point and the relationships
between stress and strain in a rigid body which obeys Hooke's law.
While part of the material covered in this chapter is a review of infor
mation generally covered in strength of materials, the subject is extended
beyond this point to a consideration of stress and strain in three dimen
sions and to an introduction to the theory of elasticity. The material
included in this chapter is important for an understanding of most of the
phenomenological aspects of mechanical metallurgy, and for this reason
it should be given careful attention by those readers to whom it is
unfamiliar. In the space available for this subject it has not been possi
ble to carry it to the point where extensive problem solving is possible.
The material covered here should, however, provide a background for
intelligent reading of the more mathematical literature in mechanical
metallurgy.
22. Description of Stress at a Point
As described in Sec. 18, it is often convenient to resolve the stresses
at a point into normal and shear components. In the general case the
shear components are at arbitrary angles to the coordinate axes, so that
it is convenient to resolve each shear stress further into two components.
The general case is shown in Fig. 21. Stresses acting normal to the
faces of the elemental cube are identified by the subscript which also
identifies the direction in which the stress acts; that is, ax is the normal
stress acting in the x direction. Since it is a normal stress, it must act
on the plane perpendicular to the x direction. By convention, values of
normal stresses greater than zero denote tension; values less than zero
17
18 Mechanical Fundamentals [Chap. 2
indicate compression. All the normal stresses shown in Fig. 21 are
tensile.
Two subscripts are needed for describing shearing stresses. The first
subscript indicates the plane in which the stress acts and the second the
direction in which the stress acts. Since a plane is most easily defined
by its normal, the first subscript refers to this normal. For example,
Tyz is the shear stress on the plane perpendicular to the y axis in the
Fig. 21 . Stresses acting on an elemental unit cube.
direction of the z axis. t„x is the shear stress on a plane normal to the
y axis in the direction of the x axis. Shearing stresses oriented in the
positive directions of the coordinate axes are positive if a tensile stress
on the same cube face is in the positive direction of the corresponding axis.
All the shear stresses shown in Fig. 21 are positive.
The notation given above is the one used by Timoshenko^ and most
American workers in the field of elasticity. The reader is reminded,
however, that several other systems of notation are in use. Before
attempting to read papers in this field it is important to become familiar
with the notation which is used.
It can be seen from Fig. 21 that nine quantities must be defined in
order to establish the state of stress at a point. They are ax, oy, cr^, Txy,
Txz, Tyx, Tyz, T^x, and Tzy. Howcvcr, some simplification is possible. If we
assume that the areas of the faces of the unit cube are small enough so
that the change in stress over the face is negligible, by taking the sum
1 S. P. Timoshenko and J. N. Goodier, "Theory of Elasticity," 2d ed., McGraw
Hill Book Company, Inc., New York, 1951.
)ec.
23]
Stress and Strain Relationships For Elastic Behavior 19
mation of the moments of the forces about the z axis it can be shown that
Tiy = Tyx.
(tjcu a?/ Az) Ax = (Tyx Ax Az) Ay
• • Txy == Tyx
and in like manner
TXZ ^^ TzX Tyz = Tgy
(21)
Thus, the state of stress at a point is completely described by six com
ponents/ three normal stresses and three shear stresses, ax, ay, cr^, Txy,
Txz, Tyz.
23. State of Stress in Two Dimensions (Plane Stress)
Many problems can be simplified by considering a twodimensional
state of stress. This condition is frequently approached in practice when
)
Fig. 22. Stress on oblique plane, two dimensions.
one of the dimensions of the body is small relative to the others. For
example, in a thin plate loaded in the plane of the plate, there will be no
stress acting perpendicular to the plate. The stress system will consist
of two normal stresses (Xx and ay and a shear stress Txy. A stress con
dition in which the stresses are zero in one of the primary directions is
called plane stress.
Figure 22 illustrates a thin plate with its thickness normal to the
plane of the paper. In order to know the state of stress at point in
the plate, we need to be able to describe the stress components at for
any orientation of the axes through the point. To do this, consider an
oblique plane normal to the plane of the paper at an angle 6 between the
^ For a more complete derivation see C. T. Wang, "Applied Elasticity," pp. 79,
McGrawHill Book Company, Inc., New York, 1953.
20 Mechanical Fundamentals [Chap. 2
X axis and the normal A^ to the plane. It is assumed that the plane
shown in Fig. 22 is an infinitesimal distance from and that the ele
ment is so small that variations in stress over the sides of the element
can be neglected. The stresses acting on the oblique plane are the normal
stress a and the shear stress r. The direction cosines between N and the
X and y axes are I and m, respectively. From the geometry of Fig. 22,
it follows that I = cos 6 and m = sin 6. If A is the area of the oblique
plane, the areas of the sides of the element perpendicular to the x and y
axes are Al and Am.
Let Sx and Sy denote the x and y components of the total stress acting
on the inclined face. By taking the summation of the forces in the x
direction and the y direction, we obtain
SxA = CFxAl f TxyA^m
SyA = ay Am + TxyAl
or Sx = (Tx cos 6 f Txy sin 6
Sy = (Ty sin 6 + Txy cos
The components of Sx and Sy in the direction of the normal stress a are
SxN = Sx COS 6 and SyN = Sy sin 9
so that the normal stress acting on the oblique plane is given by
(T = Sx COS 6 j Sy sin 6
cr = (Tx cos^ d \ (jy sin^ d \ 2Txy sin 6 cos 6 (22)
The shearing stress on the oblique plane is given by
T = Sy cos 6 — Sx sin 9
T = Txy{cos~ 6 — sin^ 9) + (oy — (Xx) sin 9 cos 6 (23)
To aid in computation, it is often convenient to express Eqs. (22) and
(23) in terms of the double angle 29. This can be done with the follow
ing identities:
, , cos 20 + 1
uuo u
2
sm2 9
_ 1
—
cos 29
2
2 sin
9 cos
9
=
sin 29
cos^ 9 —
sin^
9
=
cos 29
Equations (22) and (23) now become
O'x \ (Ty (Xx — (T
a = — ^ + ' ^ " cos 29 + Txy sin 29 (24)
sin 26 { Txy cos 29 (25)
2 ' 2
0"u — (Xx
Sec. 23]
Stress and Strain Relationships for Elastic Behavior 21
Equations (22) and (23) and their equivalents, Eqs. (24) and (25),
describe the normal stress and shear stress on any plane through a point
in a body subjected to a planestress situation. Figure 23 shows
cr^ = 2,000psi
Fig. 23. Variation of normal stress and shear stress with angle 6.
the variation of normal stress and shear stress with 6 for the biaxial
planestress situation given at the top of the figure. Note the following
important facts about this figure :
1. The maximum and minimum values of normal stress on the oblique
plane through point occur when the shear stress is zero.
2. The maximum and minimum values of both normal stress and shear
stress occur at angles which are 90° apart.
3. The maximum shear stress occurs at an angle halfway between the
maximum and minimum normal stresses.
4. The variation of normal stress and shear stress occurs in the form
of a sine wave, with a period oi 6 = 180°. These relationships are valid
for any state of stress.
22 Mechanical Fundamentals [Chap. 2
For any state of stress it is always possible to define a new coordinate
system which has axes perpendicular to the planes on which the maxi
mum normal stresses act and on which no shearing stresses act. These
planes are called the principal planes, and the stresses normal to these
planes are the principal stresses. For twodimensional plane stress there
will be two principal stresses ai and a 2 which occur at angles that are 90°
apart (Fig. 23). For the general case of stress in three dimensions there
will be three principal stresses ai, 02, and 03. According to convention,
(Ti is the algebraically greatest principal stress, while 03 is the algebraically
smallest stress. The directions of the principal stresses are the principal
axes 1, 2, and 3. Although in general the principal axes 1, 2, and 3 do
not coincide with the cartesiancoordinate axes x, y, z, for many situations
that are encountered in practice the two systems of axes coincide because
of symmetry of loading and deformation. The specification of the princi
pal stresses and their direction provides a convenient way of describing
the state of stress at a point.
Since by definition a principal plane contains no shear stress, its angu
lar relationship with respect to the xy coordinate axes can be determined
by finding the values of 6 in Eq. (23) for which r = 0.
Ti,v(cos2 6 — sin 6) + {uy — Ux) sin 6 cos =
rxy _ sin e cos d ^ >^(sin 26) ^ 1 ^^^^ ^^
(Tx — (^y cos^ 6 — sin^ 6 cos 26 2
tan 26 = ^^^^ (26)
(Tx — Cy
Since tan 26 = tan (tt  26), Eq. (26) has two roots, 61 and 62 = 61
j mr/2. These roots define two mutually perpendicular planes which
are free from shear.
Equation (24) will give the principal stresses when values of cos 26 and
sin 26 are substituted into it from Eq. (26). The values of cos 26 and
sin 26 are found from Eq. (26) by means of the Pythagorean relationships.
sin 26 = ±
cos 26 = +
[{ax  (r,)V4 + ry^^
(OX  (Ty)/2
Substituting these values into Eq. (24) results in the expression for the
Sec. 24]
Stress and Strain Relationships for Elastic Behavior 23
maximum and minimum principal stresses for a twodimensional (biaxial)
state of stress.
(7 max = O"! I
O'miD = 0'2
0"X + (Ty \ ( a, — (Jy \ 2 T'
(27)
24. Mohr's Circle of Stress — Two Dimensions
A graphical method for representing the state of stress at a point on
any oblique plane through the point was suggested by O. Mohr. Figure
24 is a Mohr's circle diagram for a twodimensional state of stress.
Fig. 24. Mohr's circle for twodimensional state of stress.
Normal stresses are plotted along the x axis, shear stresses along the
y axis. The stresses on the planes normal to the x and y axes are plotted
as points A and B. The intersection of the line AB with the x axis
determines the center of the circle. At points D and E the shear stress
is zero, so that these points represent the values of the principal stresses.
The angle between the x axis and ax is determined by angle ACD in Fig.
24
Mechanical Fundamentals
[Chap. 2
24. This angle on Mohr's circle is equal to twice the angle between ai
and the x axis on the actual stressed body.
From Fig. 24 it can be determined that
^^ = OC + CD = ^^±^^ +
OC  CE =
2
<^X + CTy
Ax — <Ty\
2 J
I 2
I ' xy
The radius of the circle is equal to
nn "^i ~ *^2
Thus, the radius of Mohr's circle is equal to the maximum shearing
stress.
<Ti — (72
+ r^
(28)
This value is given by the maximum ordinate of the circle. Note that it
acts on a plane for which 6 = x/4 (26 = 7r/2 on Mohr's circle); i.e., the
plane on which r^ax acts bisects the angle between the principal stresses.
Mohr's circle can also be used to determine the stresses acting on any
oblique plane mm. Using the convention that 6 is positive when it is
measured clockwise from the positive x axis, we find that to determine
the stresses on the oblique plane whose normal is at an angle 6 we must
advance an angle 26 from point A in the Mohr's circle. The normal and
shearing stresses on the oblique plane are given by the coordinates of
point F. The stresses on a plane perpendicular to mm would be obtained
by advancing an additional 180° on the Mohr's circle to point G. This
shows that the shearing stresses on two perpendicular planes are numeri
cally equal. It can also be seen from Fig. 24 that OF' + OG' = 20C.
Therefore, the sum of the normal stresses on two perpendicular planes is
a constant, independent of the orientation of the planes.
25. State of Stress in Three Di
mensions
The general threedimensional state of stress consists of three unequal
principal stresses acting at a point. This is called a triaxial state of stress.
If two of the three principal stresses are equal, the state of stress is known
as cylindrical, while if all three principal stresses are equal, the state of
stress is said to be hydrostatic, or spherical.
The determination of the principal stresses for a threedimensional
state of stress in terms of the stresses acting on an arbitrary cartesian
coordinate system is an extension of the method described in Sec. 23
Sec. 25]
Stress and Strain Relationships for Elastic Behavior 85
for the twodimensional case. Figure 25 represents an elemental free
body similar to that shown in Fig. 21 with a diagonal plane JKL of
area A. The plane JKL is assumed to be a principal plane cutting
through the unit cube, a is the principal stress acting normal to the
plane JKL. Let I, m, n be the direction cosines of a, that is, the cosines
Fis. 25. Stresses acting on elemental free body.
of the angles between <x and the x, y, and z axes. Since the free body in
Fig. 25 must be in equilibrium, the forces acting on each of its faces must
balance. The components of a along each of the axes are Sx, Sy, and S^.
Sx = (tI
Area KOL = Al
Sy = am
Area J OK = Am
S.
Area JOL
an
An
Taking the summation of the forces in the x direction results in
(tAI — (TxAl — TyxAm — TzxAn =
which reduces to
((7 — (Tx)l — Tyxtn — Tsxn =
Summing the forces along the other two axes results in
— Txyl + (O — Cry)m — T^yU =
— Txzl — TyzVl + (cr — (tO =
(29a)
(296)
(29c)
Equations (29) are three homogeneous linear equations in terms of I,
26 MechaHcal Fundamentals [Chap. 2
m, and n. The only solution can be obtained by setting the determinant
of the coefficients of I, m, and n equal to zero, since I, m, n cannot all eqtial
zero.
— Txtj cr — cr„ — T,
=
Solution of the determinant results in a cubic equation in a.
<J^ — (ox + 0"^ + O'z)"'" + {(TxCTy + (Ty(J2 + (X x(T z " T .^y — Ty^ " T ^^ff
— {(Jx(Ty(Tz + '^iTxyTyzTxz — (TxTy^ — (TyTxz ~ (^ zT xy) = (210)
The three roots of Eq. (210) are the three principal stresses ci, 02, and 03.
To determine the direction, with respect to the original x, y, z axes, in
which the principal stresses act, it is necessary to substitute oi, 02, and 03
each in turn into the three equations of Eq. (29). The resulting equa
tions must be solved simultaneously for I, m, and n with the help of the
auxiliary relationship P ^ m^ { n"^ = 1.
Let S be the total stress before resolution into normal and shear com
ponents and acting on a plane (not a principal plane), and let I, m, and n
be the direction cosines for the plane with respect to the three principal
axes.
S' = Sx"" + Sy^ + Sz' = ai'P + a^'m' + az'n' (211)
The normal stress a acting on this plane is given by
a = Sxl + SyM + SzTl = (7l/2 + 02^2 + (7371^ (212)
Therefore, the shearing stress acting on the same plane is given by
r^ = S^  <j^ = aiT + as^m^ { a^V  (cxiP + cr^m^ + a^n^)^
<vhich reduces to
^2 ^ (^^ _ o2)2^2m2 + (ai  asyPn^ + ((72  as)hn^n^ (213)
Values of t for the three particular sets of direction cosines listed below
are of interest because they bisect the angle between two of the three
principal axes. Therefore, they are the maximum shearing stresses or the
principal shearing stresses.
(214)
I
m
n
r
+ Vy2
+ Vy2
0'2 — <T3
''^ 2
±vy2
±Vy2
"1 Orj
T2  2
±vk
±V^
(X\ — (T%
Tz — o
Sec. 26]
Stress and Strain Relationships for Elastic Behavior 27
Since according to convention ai is the algebraically greatest principal
normal stress and 03 is the algebraically smallest principal stress, n has
the largest value of shear stress and it is called the maximum shear stress
ci — O'S
(215)
The maximum shear stress is important in theories of yielding and metal
forming operations. Figure 26 shows the planes of the principal shear
0\
_ — >
•mnv ~ *? "~
_ tr,  cTi
oi
. I
^2
*
>
'az
g'z ^3
Fig. 26. Planes of principal shear stresses.
stresses for a cube whose faces are the principal planes. Note that for
each pair of principal stresses there are two planes of principal shear stress,
which bisect the principal directions for the normal stresses.
26. Mohr's Circle — Three Dimensions
The discussion given in Sec. 24 of the representation of a twodimen
sional state of stress by means of Mohr's circle can be extended to three
dimensions. Figure 27 shows how a triaxial state of stress, defined by
the three principal stresses, can be represented by three Mohr's circles.
28
Mechanical Fundamentals
[Chap. 2
It can be shown ^ that all possible stress conditions within the body fall
within the shaded area between the circles in Fig. 27.
k^2
/ f
Fig. 27. Mohr's circle representation of a threedimensional state of stress.
While the only physical significance of Mohr's circle is that it gives a
geometrical representation of the equations that express the transforma
tion of stress components to different sets of axes, it is a very convenient
way of visualizing the state of stress. Figure 28 shows Mohr's circle
for a number of common states of stress. Note that the application of
a tensile stress ai at right angles to an existing tensile stress ax (Fig. 28c)
results in a decrease in the principal shear stress on two of the three sets
of planes on which a principal shear stress acts. However, the maximum
shear stress is not decreased from what it would be for uniaxial tension,
although if only the twodimensional Mohr's circle had been used, this
would not have been apparent. If a tensile stress is applied in the third
principal direction (Fig. 28d), the maximum shear stress is reduced
appreciably. For the limiting case of equal triaxial tension (hydrostatic
tension) Mohr's circle reduces to a point, and there are no shear stresses
acting on any plane in the body. The effectiveness of biaxial and tri
axialtension stresses in reducing the shear stresses results in a consider
able decrease in the ductility of the material, because plastic deformation
is produced by shear stresses. Thus, brittle fracture is invariably associ
ated with triaxial stresses developed at a notch or stress raiser. Hovf
ever. Fig. 28e shows that, if compressive stresses are applied lateral to a
tensile stress, the maximum shear stress is larger than for the case of
1 A. Nadai, "Theory of Flow and Fracture of Solids," 2d ed., pp. 9698, McGraw
Hill Book Company, Inc., New York, 1950.
Sec. 26]
Stress and Strain Relationships for Elastic Behavior 29
,2
(C)
cr, = 2 cr.
{c]
7max = r2=r, c^jj^
33.
^mox ~ ^2 ~ ^3
cr, = 202 = 203
U)
Fig. 28. Mohr's circles (threedimensional) for various states of stress, (a) Uniaxial
tension; (6) uniaxial compression; (c) biaxial tension; (d) triaxial tension (unequal);
(e) uniaxial tension plus biaxial compression.
either uniaxial tension or compression. Because of the high value of
shear stress relative to the applied tensile stress the material has an
excellent opportunity to deform plastically without fracturing under this
state of stress. Important use is made of this fact in the plastic working
of metals. For example, greater ductility is obtained in drawing wire
through a die than in simple uniaxial tension because the reaction of the
metal with the die will produce lateral compressive stresses.
30
Mechanical Fundamentals
[Chap. 2
An important state of stress is pure shear. Figure 29a illustrates
Mohr's circle for a twodimensional state of pure shear. This state of
stress is readily obtained by twisting a cylindrical bar in torsion. The
Mohr's circle for this state of stress shows that the maximum and mini
mum normal stresses are equal to the shear stress and occur at 45° to the
'/x
im
'//
K ^min~ '//
'//C
max" '//
(a)
cr^ay =cr,
Kb)
Fig. 29. Equivalent pureshear conditions, (a) Pure shear (plane stress); (6) equal
biaxial tension and compression.
shear stresses. The maximum shear stress is equal to the applied shear
stress Txii, but it occurs only on the set of planes parallel to the z axis.
On the other two sets of planes the principal shear stress is rxi//2. Note
that for threedimensional pure shear two out of the three sets of shear
planes have a value of Tn,ax = ci. An identical state of stress to pure
shear can be obtained when equal tensile and compressive stresses are
applied to a unit cube (Fig. 296). Once again Xmax = cri, but to obtain
complete identity with a state of twodimensional pure shear, the axes
must be rotated 45° in space or 90° on the Mohr's circle.
Sec. 27]
Stress and Strain Relationships for Elastic Behavior 31
27. Description of Strain at a Point
The brief description of linear strain and shear strain given in Sec. 19
can be expanded into a more generaHzed description of the strain at a
point in a rigid body. For simpUcity in illustration, Fig. 210 considers
dy ^
'^Ty'^y
dy
M
I I
I L I
.__^^
dx
.*%,i.
dx
Fig. 210. Components of strain for plane strain.
a twodimensional, or planestrain, condition where all the deformation is
confined to the xy plane. It is, however, a simple matter to generalize
the relationship obtained from this figure to the threedimensional case.
Let the coordinates of a point in an unstrained rigid body be defined by
X, y, and z. After strain is applied, the point will undergo displacements
u, V, and w in the directions x, y, and z. In order that the displacement
of the entire body be geometrically compatible, it is necessary that no
two particles occupy the same point in space or that no voids be created
within the body. In order to satisfy these requirements, the displace
ment components u, v, and w must vary continuously from point to point.
This can be accomplished if their gradients with respect to x, y, and z
have no discontinuities, and therefore the partial derivatives of u, v, and
w with respect to x, y, and z enter into the analysis.
In Fig. 210 for the planestrain condition, it can be seen that the
x component of the displacement of K to K' is the displacement of J,
given by u, plus the rate of change of u along the distance dx, given by
(du/dx) dx. The unit linear strain in the x direction, e^, is given by
_ J'K' — JK _ [dx \ (du/dx) dx] — dx _ du /q ir ^
^^ ~ JK ~ dx " di ^^^ ^^
Similarly
J'M'  JM _ dy \ (dv/dy) dy  dy _ dv
JM
dy
dy
(2166)
32 Mechanical Fundamentals [Chap. 2
and if the z direction were being considered,
e. = ^ (216c)
The shearing strain 7^^ at J' is given by the change in angle of the two
elements originally parallel to the x and y axes.
7x^ = /KJM  LK'J'W = IBJ'M' + lAJ'K'
Since, for the small strains involved, the tangent of the angle equals the
angle, from Fig. 210 it can be seen that
(du/dy) dy (dv/dx) dx du dv /n icj\
^ = dy + dx ^di^di ^^^^^^
In the same way, it can be shown that
du dw /o ic \
^ ^Tz^Tx (216e)
dy dw
Thus, six terms are necessary completely to describe the state of strain
at a point, e^, ey, e^, 7^^, 75^^, 7x2.
In complete analogy with stress, it is possible to define a system of
coordinate axes along which there are no shear strains. These axes are
the principal strain axes. For an isotropic body the directions of the
principal strains coincide with those of the principal stresses.^ An ele
ment oriented along one of these principal axes will undergo pure exten
sion or contraction without any rotation or shearing strains. As with
principal stresses, the largest and smallest linear strains at a point in the
body are given by the values of the principal strains.
Methods similar to those used in Sec. 25, for the derivation of the
equation for the values of the three principal stresses, and in Sec. 23,
for the normal and shear stresses on any plane through a point in the
body, can be used for the derivation^ of similar quantities in terms of
strain. However, these equations may be obtained much more easily by
replacing a and r in the stress equation by e and 7/2. For example, the
linear strain on any plane in a twodimensional situation can, from Eq.
(22), be expressed by
es = ex cos^ 6 \ Cy sin^ 6 \ 72y sin 6 cos d (217)
The three principal strains ei > 62 > 63 are the three roots of the follow
' For a derivation of this point see Wang, op. cii., pp. 2627.
2 Timoshenko and Goodier, op. cit., pp. 221227.
Sec. 28] Stress and Strain Relationships For Elastic Behavior 33
ing equation [obtained from Eq. (210)] :
e^  (fx + ey \ e,)e'^ + [e^Cy + e^e, + e^e,  }^i(yl,, + tL + ylz)]e
 e^eye, + Mi^xll^ + e,,7L + ^ztL  lx,nxzi,z) = (218)
By following through with this analogy, the equations for the principal
shearing strains can be obtained from Eq. (214).
7i = €"2 — f's
Tmax = 72 = fl  63 (219)
73 = ei — 62
The volume strain, or cubical dilatation, is the change in volume per
unit of original volume. Consider a rectangular parallelepiped with edges
dx, dy, dz. The volume in the strained condition is
(1 + ei)(l + 62) (1 + 63) dx dy dz
From its definition, the volume strain A is given by
_ (1 + 61) (1 + 62) (1 + 63) dx dy dz — dx dy dz
" dx dy dz
= (1 f ei)(l +e2)(l +63)  1
which for small strains, after neglecting the products of strains, becomes
A = ei 4 62 + 63 (220)
28. Measurement of Surface Strain
Except in a few cases involving contact stresses, it is not possible to
measure stress directly. Therefore, experimental measurements of stress
are actually based on measured strains and are converted to stresses by
means of Hooke's law and the more general relationships which are given
in Sec. 210. The most universal strainmeasuring device is the bonded
wire resistance gage, frequently called the SR4 strain gage. These gages
are made up of several loops of fine wire or foil of special composition,
which are bonded to the surface of the body to be studied. When the
body is deformed, the wires in the gage are strained and their electrical
resistance is altered. The change in resistance, which is proportional to
strain, can be accurately determined with a simple Wheatstonebridge
circuit. The high sensitivity, stability, comparative ruggedness, and ease
of application make resistance strain gages a very powerful tool for strain
determination.
For practical problems of experimental stress analysis it is often impor
tant to determine the principal stresses. If the principal directions are
known, gages can be oriented in these directions and the principal stresses
34
Mechanical Fundamentals
[Chap. 2
\a} id]
Fig. 211. Typical straingage rosettes, (a) Rectangular; (6) delta.
r
2
c
c
1
'^2
D
b t
oc\.
z
V
i
y
0\2^ \
I
< — ^ — *■
B
^b "
Cq,
Fig. 21 2. Mohr's circle for determination of principal strains.
determined quite readily. In the general case the direction of the princi
pal strains will not be known, so that it will be necessary to determine
the orientation and magnitude of the principal strains from the measured
strains in arbitrary directions. Because no stress can act perpendicular
to a free surface, straingage measurements involve a twodimensional
state of strain. The state of strain is completely determined if e^, e^,
and ^xM can be measured. However, strain gages can make only direct
Sec. 29] Stress and Strain Relationships for Elastic Behavior 35
readings ©f linear strain, while shear strains must be determined indirectly.
Therefore, it is the usual practice to use three strain gages separated at
fixed angles in the form of a "rosette," as in Fig. 211. Straingage read
ings at three values of 6 will give three simultaneous equations similar to
Eq. (217) which can be solved for e^, Cy, and yxy The twodimensional
version of Eq. (218) can then be used to determine the principal strains.
Equations for directly converting straingage readings into principal
stresses for the two rosettes shown in Fig. 211 will be found in Table 22.
A more convenient method of determining the principal strains from
straingage readings than the solution of three simultaneous equations in
three unknowns is the use of Mohr's circle. In constructing a Mohr's
circle representation of strain, values of linear normal strain e are plotted
along the x axis, and the shear strain divided by 2 is plotted along the
y axis. Figure 212 shows the Mohr's circle construction^ for the gener
alized straingage rosette illustrated at the top of the figure. Straingage
readings Ca, Cb, and Cc are available for three gages situated at arbitrary
angles a and /3. The objective is to determine the magnitude and orien
tation of the principal strains ei and ez.
1. Along an arbitrary axis X'X' lay off vertical lines aa, bb, and cc
corresponding to the strains Ca, Cb, and Cc.
2. From any ])oint on the line bb (middle strain gage) draw a line
DA at an angle a with bb and intersecting aa at point A. In the same
way, lay off DC intersecting cc at point C.
3. Construct a circle through A, C, and D. The center of this circle is
at 0, determined by the intersection of the perpendicular bisectors to
CD and AD.
4. Points A, B, and C on the circle give the values of e and 7/2 (meas
ured from the new x axis through 0) for the three gages.
5. Values of the principal strains are determined by the intersection of
the circle with the new x axis through 0. The angular relationship of ei
to the gage a is onehalf the angle AOP on the Mohr's circle {AOP = 26).
29. StressStrain Relations
In the first chapter it was shown that uniaxial stress is related to uni
axial strain by means of the modulus of elasticity. This is Hooke's law
in its simplest form,
(T. = Ee, (221)
where E is the modulus of elasticity in tension or compression. While a
tensile force in the x direction produces a linear strain along that axis,
» G. Murphy, /. Appl. Mech., vol. 12, p. A209, 1945; F. A. McClintock, Proc. Soc.
Exptl. Stress Analysis, vol. 9, p. 209, 1951.
36 Mechanical Fundamentals [Chap. 2
it also produces a contraction in the transverse y and z directions. The
ratio of the strain in the transverse direction to the strain in the longi
tudinal direction is known as Poisson's ratio, denoted by the symbol v.
Cy == e^ = PC:, =  ^ (222)
Only the absolute value of v is used in calculations. Poisson's ratio is
0.25 for a perfectly isotropic elastic material, but for most metals the
values^ are closer to 0.33.
The generalized description of Hooke's law says that for a body acted
upon by a general stress system the strain along any principal axis is due
to the stress acting along that axis plus the superimposed strain resulting
from the Poisson effect of the principal stresses acting along the other
two axes.
1 V V
= ^ [ol  J'(0"2 + org)]
and for the other two principal axes
^2 = p, [o2 — v((Ti + 03)]
^ (223)
^3 = p, ks — v((ri + 02)]
For a biaxial state of plane stress, 03 = 0, and Eqs. (223) reduce to
ei = ^ (oi  W2)
^2 = p, (o"2 — vari) {223a>
63 = ~ E^'^^ ~^ ^^^
Note that, even when the stress along the third axis is zero, the strain
along that axis is not zero (unless ci = —02).
In the case of plane strain, es = 0, and the strainstress relationships
become
ei = — p^ [(1  j/)(Ti  W2]
1 + . (22^^^
62 = — p — [(1 — v)<r2 — wi]
' A derivation which intuitively suggests that v = 0.33 has been presented by F. R.
Shanley, "Strength of Materials," pp. 138139, McGrawHill Book Company, Inc.,
New York, 1957.
Sec. 29]
Stress and Strain Relationships for Elastic Behavior 37
Shear stress is related to shear strain by relationships similar to Eq.
(221).
Txy = Gy^y T„ = Gy^^ Ty^ = Gyy: (224)
The proportionality constant G is the modulus of elasticity in shear, or the
modulus of rigidity.
Three material constants E, G, and v are involved in the description
of the elastic behavior of materials. It will be shown that these con
stants are related, so that for an isotropic material there are only two
1 1 1
1 1 1
— *■
^
a
y
tiz
— *
— >
^ \\ °
c
J i J
\ \ \
(
a)
Fig. 213. (a) Element subjected to pure shear; (6) stresses on triangle Oab before
deformation; (c) after deformation.
independent elastic constants. Consider a rectangular element which is
subjected to a condition of pure shear (Fig. 213) (see the Mohr's circle
in Fig. 296). <Tx and ay are principal stresses since no shear stresses are
present on the faces on which they act. The maximum shearing stress
will lie on a 45° plane, and if ay = —dx, the value of Xmax will be (Ty. The
shearing stresses distort the element in the manner shown in Fig. 21 3c.
Oa is elongated, Ob is shortened, and ab does not change in length. Angle
Oab is reduced by an amount 7/2.
. /w y\ Ob' 1
+ ex
1 +
Since, for elastic strains, 7 is a small angle
tan (x/4)  tan (7/2)
/tt _ 7\ ^ J
\4 V 1
tan I  —
_ 1  7/2
H tan (7r/4) tan (7/2) 1 \ 7/2
For pure shear
— (Tx = T
38 Mechanical Fundamentals
Substitution into Eq. (223) produces
[Chap. 2
_ _ _ 1 + V _ ^ + V
^V ^X ^V n *~~ ^
E
E
Equating the two expressions for tan (7r/4 — 7/2) and substituting the
above equation gives
Comparing this relation with the generalized form of Eq. (224) results in
the general relationship between the three elastic constants. Typical
values for a number of materials are given in Table 21.
G =
E
2(1 + v)
(226)
Table 21
Typical Roomtemperature Values of Elastic Constants
FOR Isotropic Materials
Material
Aluminum alloys
Copper
Steel (plain carbon and lowalloy) .
Stainless steel (188)
Titanium
Tungsten
Modulus of
elasticity,
10e psi
10.5
16.0
29.0
28.0
17.0
58.0
Shear
modulus,
106 psi
4.0
6.0
11.0
9.5
6.5
22.8
Poisson'
ratio
0.31
0.33
0.33
0.28
0.31
0.27
The addition of the three equations giving the strain produced by the
three principal stresses [Eq. (223)] results in
ei + 62 + 63 = — ^ — (oi + 02 + o's)
(227)
The lefthand term of Eq. (227) is the volume strain A. Rewriting Eq.
(227) results in
S'^™ (1  2p)
A
E
(228)
where am is the average of the three principal stresses. This can be
written as
_arn _ E^
" ~ A ~ 3(1  2v)
(229)
The constant k is the volumetric modulus of elasticity, or the hulk modulus.
Sec. 210] Stress and Strain Relationships for Elastic Behavior 39
The bulk modulus is therefore the ratio of the average stress to the vol
ume strain. It is frequently evaluated for conditions of hydrostatic com
pression, where o„, is equal to the hydrostatic pressure.
210. Calculation of Stresses from Elastic Strains
The general equations expressing stress in terms of strain are consider
ably more complicated than the equations giving strain in terms of stress
[Eqs. (223)]. The simultaneous solution of these equations for oi, ai, and
(73 results in
_ vE . . E
'''  (1 + .)(!  2u) ^ + TlTv ''
= (l + .)a2.) ^ + lf/ (230)
vE ^ , E
"' ~ 0"T".)(i  2.) ^ + IT". ''
where A = ei { e2 \ e^. The term X = vE/(l \ p)(l — 2v) is called
Lame's constant. By using this constant the above equations can be
simplified to
(71 = XA f 2Gei
(72 = XA f 2Ge2 (231)
(73 = XA + 2Ge3
For the case of plane stress (a^ = 0) two simple and useful equations
relating the stress to the strains may be obtained by solving the first two
of Eqs. (223) simultaneously for ai and (72.
ci = :j^^ — I (ei + 1^62)
E ' (232)
02 = 1 5 (^2 + vei)
These equations can be very useful for determining the values of princi
pal stress from straingage measurements. Note that even when a strain
gage is oriented in the principal direction it is not correct to multiply the
strain by the elastic modulus to get the stress in that direction. Because
of the Poisson effect, corrections for lateral strains must be made by using
Eqs. (232).
Usually the methods described in Sec. 28 will have to be employed to
determine the values of principal strain before these equations can be
used. A shortcut procedure is to use a straingage rosette with three
gages at fixed orientations for which relationships have been worked out
between the strains measured by each gage and the principal stresses.
Table 22 gives the relationships between the principal stresses and the
iH m
t ^
Eh HI
o
o >
CC
m .2
(M S "If
I 03 &
=^ H C
Z si
n
CO
P5
0^
+
+
+
+ +
6q IcN &q
tf
+
+
>
+
^
+
1
T— 1
^
;^
+
1
+
^H
bqicN Ki
P4
+
+
>
+
K5
+
&q
+
3
III
^
«t>
^
1
lU
•ij
+
^ ^x
,^
e ^
1
^ ^
t I (M
+
+
1
T
I I (M
^
40
Sec. 211] Stress and Strain Relationships for Elastic Behavior 41
strain readings €«, ei>, and Cc for the three gages in the rectangular and
delta rosettes shown in Fig. 211. The derivation of the equations and
graphical solutions for these equations are discussed in texts^ on strain
gage technology.
21 1 . Generalized StressStrain Relationships
The stressstrain relationships given in the previous two sections con
tain three elastic constants E, G, and v. These are the only material
constants needed to describe the elastic stressstrain behavior provided
that the material can be considered isotropic. However, many materials
are anisotropic ; i.e., the elastic properties vary with direction. A metallic
single crystal is an extreme example of an anisotropic elastic material,
while coldworked metals may also exhibit anisotropic elastic behavior.
However, in the usual case with engineering materials the grain size is
small enough and the grains are in a sufficiently random arrangement
so that the equations based on isotropic conditions may be used.
In order to consider the possibility of elastic constants which vary with
orientation in the material, Hooke's law can be written in completely
general terms as a linear relationship between strain and stress.
^x = SllCTx + Sl2(^y f Sn(72 f SuTxy + SnTyz \ SieTzx
€y = S2lO'x + S^lCy + S230'z + SiiTxy + S^bTyz + 'S26Tzi
Cz = SsiCTx + S320^y + S^^a^ + »Sl34Txy + S^bTyz + Ss^Tzx /o oo\
7x2, = Suffjc + Si2Cry + *S430'j j SuTxy + Si^Tyz + S^^Tsx
Jyz = SblOx + Sh20'y + Sb^O'z \ SbiTxy + Sf,5Tyz + S^GTzx
7zx = SeiCx + S62cry + ♦SeaO'z + S^iTxy + SebTyz + SeeTzx
The constants ^Sn, *Si2, . . . , >S,y are the elastic compliances. Note that
these equations indicate that a shear stress can produce a linear strain
in an elastically anisotropic material. A similar set of six equations
relates the stress to the strain in terms of the elastic coefficients Cn,
Cl2, ■ • ■ , Cij.
cTx = Ciiex + CuCy \ Ci^ez j Ciaxy + Cibyyz + Ci&yzx
(234)
Txy = C ilCx + CiiCy + Ci^Cz \ Cnyxy + Cihjyz + C*46T2X
Thus, in order to calculate the stress from the strain in the most
general circumstances, it is necessary to know 6 strain components and
36 elastic coefficients. Fortunately, symmetry considerations can reduce
considerably the number of necessary constants. The elastic constants
1 C. C. Perry and H. R. Lissner, "The Strain Gage Primer," McGrawHill Book
Company, Inc., New York, 1955.
42
Mechanical Fundamentals
[Chap. 2
with unequal subscripts are equivalent when the order of the subscripts
is reversed.
Sij — Sji
Ci
\jji
Therefore, even for the least symmetrical crystal structure (triclinic) the
number of independent constants is reduced to 21. For metals, which
all have crystal structures of relatively high symmetry, the maximum
number of constants that need to be considered is 12. Thus, Eqs. (233)
can be written as
ex = SnCx + Su(Ty + Snf^z Jxp = SuTxy
62 = Si\(^x + Sz2(^v + SzZ(^z Izx — SeeTzx
(235)
By comparing these equations with Eqs. (223) and (224) it can be seen
that *Sii is the reciprocal of the modulus of elasticity in the x direction,
that S21 determines the component of linear strain produced in the y
Table 23
Elastic Compliances for Metal Crystals
Units of 10~^2 cm^/dyne
Metal
Aluminum
Copper
Iron
Iron (polycrystalline)
Tungsten
Magnesium
Zinc
Sii
1.59
1.49
0.80
0.48
0.26
2.23
0.82
^12
0.58
0.62
0.28
0.14
0.07
0.77
+0.11
s.
3.52
1.33
0.80
1.24
0.66
6.03
2.50
013
012
012
012
012
0.49
0.66
S3
Sn
Su
Su
Sn
Sn
1.98
2.64
direction due to ax, equivalent to v/E, and that Szi represents the same
thing for the z direction. Also, the compliance S^i is the reciprocal of
the shear modulus.
For metals which exist in one of the cubic crystal structures
'Jll = *J22 = AJ33, *J12 = *J13 = S2I = 023 = Ssi = 032, and O44 — *^55 = 066.
Therefore, Eq. (235) may be written as
Cx = Siiax + >Si2(o"j^ + (Tz) 7x2/ = SuTxy
Qy = Snay + *Sl2(0'2 + <Tx) lyz — SiiTyz
62 = 8\\Crz + *Sl2(o'x f (Ty) Izx = & \^T zx
(236)
The identity between these equations and Eqs. (223) and (224) is
apparent. Therefore, for a material with a cubic crystal structure there
are three independent elastic constants, while for a truly isotropic mate
rial there are only two independent elastic constants. For an isotropic
Sec. 212] Stress and Strain Relationships for Elastic Behavior 43
material the relationship between these constants is given by
044 = 2(oii — 012)
Table 23 lists some typical values for elastic compliance. By applying
the above relationship compare the difference in isotropy between copper
and tungsten.
21 2. Theory of Elasticity
The mathematic theory of elasticity entails a more detailed consider
ation of the stresses and strains in a loaded member than is required by
icr.^^Bdy)dx
dy
dTxy
aj^dy^
Tjrydyy
dy
dx
>^^'^/ dy
dy) dy
*(o>+^<i^)^/
TxydX
ay dx
Fig. 214. Forces acting on volume element.
the ordinary methods of analysis of strength of materials. The solutions
arrived at by strength of materials are usually made mathematically
simpler by the assumption of a strain distribution in the loaded member
which satisfies the physical situation but may not be mathematically
rigorous. In the theory of elasticity no simplifying assumptions are made
concerning the strain distribution.
As with strength of materials, the first requirement for a solution is to
satisfy the conditions of equilibrium. Figure 214 illustrates the forces
acting on an element of the body for a planestress situation. Taking the
summation of forces in the x and ij directions results in
2P. =
dx
+
dTru
=
^^• = l"+^"+«°
(237)
The term pg arises from the consideration of the weight of the body,
where p is the mass per unit volume and g is the acceleration of gravity.
The above equations constitute the equations of equilibrium for plane
44 Mechanical Fundamentals [Chap. 2
stress. For a threedimensional stress system there will be three equa
tions,^ each containing one partial derivative of the normal stress and
two partial derivatives of the shear stresses.
Equations (237) must be satisfied at all points throughout the body.
Note that these equilibrium equations do not give a relationship between
the stresses and the external loads. Instead, they give the rate of change
of the stresses at any point in the body. However, the relationship
between stress and external load must be such that at the boundary of
the body the stresses become equal to the surface forces per unit area;
i.e., it must satisfy the boundary conditions.
One of the important requirements of the theory of elasticity is that
the deformation of each element must be such that elastic continuity is
preserved. Physically, this means that the stresses must vary so that
no gaps occur in the material. The equations of compatibility for the
twodimensional case can be obtained from the definitions for strain in
terms of the displacements u and v (Sec. 27).
du . .
'^ = Ty ^^^
du , dv / V
Txy = T + — (C)
dy dx
These three equations show that there is a definite relationship between
the three strains at a point, because they are expressed in terms of two
displacements u and v. Differentiating Eq. (a) twice with respect to y,
Eq. (h) twice with respect to x, and Eq. (c) with respect to both x and y
results in
dy^ ax ox dy
Equation (238) is the equation of compatibility in two dimensions. If
the strains satisfy this equation, they are compatible with each other
and the continuity of the body is preserved. The equation of compati
bility can be expressed in terms of stress by differentiating Eqs. (223)
and (224) and substituting into Eq. (238).
W  'W' ^ ^' ' "^' ^ ^ '^ '^ didy ^^^^^
If values of o^, ay, and r^cy satisfy Eq. (239), it can be considered that
the strains which accompany these stresses are compatible and that the
continuity of the body will be preserved.
^ Timoshenko and Goodier, op. cit., chap. 9
Sec. 212] Stress and Strain Relationships for Elastic Behavior 45
Basically, the solution of a problem with the theory of elasticity
requires the determination of an expression for the stresses (Tx, <Ty, r^y in
terms of the external loads that satisfies the equations of equilibrium
[Eqs. (237)], the equations of compatibility [Eq. (239)], and the bound
ary conditions. Such a solution generally involves considerable mathe
matical agility. Most of the complications attending the theory of
elasticity arise out of the necessity for satisfying the requirement of
continuity of elastic deformation. In solutions by strength of materials
continuity of deformation is not always satisfied. Continuity is main
tained by local plastic yielding. Generally, this does not have an impor
tant effect on the solution because the effects of yielding do not extend
far beyond the region where they occur. In other problems, such as the
determination of stresses at geometrical discontinuities (stress raisers),
localized yielding is important, and the methods of the theory of elasticity
must be used.
One method which is used for the solution of problems by the theory
of elasticity is to find a function $ in x and y which satisfies Eqs. (237)
and (239) and also expresses the stresses in terms of the loads. Such a
function is usually called the Airy stress function. Airy^ showed that
there will always exist a function in x and y from which the stresses at
any point can be determined by the following equations:
^ = ^  pgy ^^ = ^. m ^^y   ^^j (240)
Equations (240) satisfy the equations of equilibrium. In order to satisfy
the compatibility equation, Eqs. (240) are differentiated and substituted
into Eq. (239).
dx* " 3.12 dy' "^ dy
4 + 2— ,f— , = (241)
If a stress function can be found for the problem which satisfies Eq.
(241), the stresses are given by Eqs. (240) provided that Eqs. (240)
also satisfy the boundary conditions of the problem. The discovery of
a stress function which satisfies both Eq. (241) and the boundary con
ditions is usually not easy, and therefore only a fairly limited number of
problems have been solved by this technique. For problems with com
plicated geometry and loading systems it is usually necessary to use
finitedifference equations and relaxation methods for the solution of the
problem.
1 G. B. Airy, Brit. Assoc. Advance, Sci. Rept., 1862.
^ R. V. Southwell, "Relaxation Methods in Theoretical Physics," Oxford University
Press, New York, 1946.
46
Mechanical Fundamentals
[Chap. 2
213. Stress Concentration
A geometrical discontinuity in a body, such as a hole or a notch,
results in a nonuniform stress distribution at the vicinity of the discon
tinuity. At some region near the discontinuity the stress will be higher
than the average stress at distances removed from the discontinuity.
Thus, a stress concentration occurs at the discontinuity, or stress raiser.
Figure 215a shows a plate containing a circular hole which is subjected
t t
LA.
~fd
\ \ \ \
{a) [b]
Fig. 215. Stress distributions due to (a) circular hole and (6) elliptical hole.
to a uniaxial load. If the hole were not present, the stress would be
uniformly distributed over the cross section of the plate and it would be
equal to the load divided by the crosssectional area of the plate. With
the hole present, the distribution is such that the axial stress reaches a
high value at the edges of the hole and drops off rapidly with distance
away from the hole.
The stress concentration is expressed by a theoretical stressconcen
tration factor Kt. Generally Kt is described as the ratio of the maximum
stress to the nominal stress based on the net section, although some
workers use a value of nominal stress based on the entire cross section
of the member in a region where there is no stress concentrator.
Kt =
^nominal
(242)
In addition to producing a stress concentration, a notch also creates a
localized condition of biaxial or triaxial stress. For example, for the
circular hole in a plate subjected to an axial load, a radial stress is pro
duced as well as a longitudinal stress. From elastic analysis,^ the stresses
produced in an infinitely wide plate containing a circular hole and axially
^ Timoshenko and Goodier, oj). cit. pp. 7881.
Sec. 21 3] Stress and Strain Relationships for Elastic Behavior 47
loaded can be expressed as
Examination of these equations shows that the maximum stress occurs
at point A when Q = 7r/2 and r = a. For this case
(Te = 3(7 = (Tmax (244)
where a is the uniform tensile stress applied at the ends of the plate.
The theoretical stressconcentration factor for a plate with a circular hole
is therefore equal to 3. Further study of these equations shows that
(To = —a for r — a and = 0. Therefore, when a tensile stress is applied
to the plate, a compressive stress of equal magnitude exists at the edge
of the hole at point B in a direction perpendicular to the axis of loading
in the plane of the plate.
Another interesting case for which an analytical solution for the stress
concentration is available^ is the case of a small elliptical hole in a plate.
Figure 2156 shows the geometry of the hole. The maximum stress at
the ends of the hole is given by the equation
K^ + ^O
(245)
Note that, for a circular hole (a = b), the above equation reduces to
Eq. (244). Equation (245) shows that the stress increases with the
ratio a/b. Therefore, a very narrow hole, such as a crack, normal to the
tensile direction will result in a very high stress concentration.
Mathematical complexities prevent the calculation of elastic stress
concentration factors in all but the simplest geometrical cases. Much of
this work has been compiled by Neuber,^ who has made calculations for
various types of notches. Stressconcentration factors for practical prob
lems are usually determined by experimental methods.^ Photoelastic
analysis^ of models is the most widely used technique. This method is
especially applicable to planestress problems, although it is possible to
make threedimensional photoelastic analyses. Figure 216 shows typical
1 C. E. Inglis, Trans. Inst. Naval Architects, pt. 1, pp. 219230, 1913.
2 H. Neuber, "Theory of Notch Stresses," Enghsh translation, J. W. Edwards,
Publisher, Inc., Ann Arbor, Mich., 1946.
'' M. Hetenyi, "Handbook on Experimental Stress Analysis," John Wiley & Sons,
Inc., New York, 1950.
* M. M. Frocht, "Photoelasticity," John Wiley & Sons, Inc., New York, 1951.
48
Mechanical Fundamentals
[a.ap. 2
t
6'
(a)
0.2 0.4 0.6 0.8 1.0
P\
*l 2r 1
o.o
3.4
1
b
V
^ r
3.0
\
\ \
2.6
\
\
b
,\
\^
r
2.2
\ N
A
\
V
\
1.8
^^^
^j>^
"■"* — _^
1.4
i) 1
/
■~~
■*=*»
■ .
==
r 2
1
1 1
{b)
0.2
0.4
r/h
0.6
0.8
■^
iijijji
3.2
3.0
•2.8
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
I
/
•^ =i'.^^
^
\
= ;.«?5
\
\ ^
^h
V
A
\
\
\
/^
^
i^^
\
>
^
<J
^
>c)
0.2 0.4 0.6 0.8 1.0
Fig. 216. Theoretical stressconcentration factors for different geometrical
Sec. 213]
Stress and Strain Relationships for Elastic Behavior 49
?.^
CO
2.2
1.8
\
\
1.4
\
.^
1.0
"■^
' —
—
—
id)
0.1 0.2 0.3 0.4 0.5
r/h
I I
Mr
I 1 1
\Ma
FF^R
^3.0
£2.8
o
2.6
o
\^A
£2.2
§2.0
I 1.8
^ 1.6
\
4
>
_^^
^
\
<; —
>,—
(e)
0.10 0.20
0.30
— ^
Mr
3.4
^ 3.0
o
o
f2.6
.o
12.2
c
o
§ 18
I 1.4
en
1.0
(
\\
1
\
■^ = 2.00
\
V'
1 \
\
\ '
A
^1.20
\
y
(^
\
\
\
\
\
\
LI ' '
—
\
\
%
■^
\
\
7^
—
^
s
—
^^
^
■""
i_
0.04 0.08
r/d
0.12
shapes. {Ajler G. H. Neugehauer, Product Eng. vol. 14, pp. 8287, 1943.)
50 Mechanical Fundamentals [Chap. 2
curves for the theoretical stressconcentration factor of certain machine
elements that were obtained by photoelastic methods. Much of the
information on stress concentrations in machine parts has been collected
by Peterson. 1
The effect of a stress raiser is much more pronounced in a brittle
material than in a ductile material. In a ductile material, plastic defor
mation occurs when the yield stress is exceeded at the point of maximum
stress. Further increase in load produces a local increase in strain at the
critically stressed region with little increase in stress. Because of strain
hardening, the stress increases in regions adjacent to the stress raiser,
until, if the material is sufficiently ductile, the stress distribution becomes
essentially uniform. Thus, a ductile metal loaded statically will not
develop the full theoretical stressconcentration factor. However, redis
tribution of stress will not occur to any extent in a brittle material, and
therefore a stress concentration of close to the theoretical value will
result. Although stress raisers are not usually dangerous in ductile
materials subjected to static loads, appreciable stressconcentration
effects will occur in ductile materials under fatigue conditions of alter
nating stresses. Stress raisers are very important in the fatigue failure
of metals and will be discussed further in Chap. 12.
214. Spherical and Deviator Components of Stress and Strain
Experiment shows that materials can withstand very large hydrostatic
pressures (spherical state of stress) without undergoing plastic defor
mation. In many problems, particularly in the theory of plasticity, it is
desirable to designate the part of the total stress which can be effective
in producing plastic deformation. This is known as the stress deviator a'.
The other component is the spherical, or hydrostatic, component of stress c" .
a" = "' + ""' + "' ^ p (246)
The stress deviator is given by the following equations:
, ,, 2c7i — (72 — Cs
O"! = 01 — (Ti =
' I' 2 o2 — 01—03 /9 A^\
(J, = 02 — 02 = o K^^n
, ,, 2o3 — 01 — 02
0^3 = 03 — 0"3 ^
It can be readily shown that (j\ + 03 + o's = 0.
1 R. E. Peterson, "Stressconcentration Design Factors," John Wiley & Sons, Inc.,
New York, 1953.
3
2ei
 62 
 63
3
2^2
 ei 
 63
3
263
 ei 
 62
Sec. 214] Stress and Strain Relationships for Elastic Behavior 51
The principal stress direction of the stress deviator is the same as that
of the principal stress of the total stress. Thus, (t[, has the same direc
tion as a I. Since an incompressible isotropic body is not deformed by
hydrostatic pressure, the deformation depends entirely on the stress
deviator, the spherical component making no contribution.
In a completely analogous manner the strain at a point can be divided
into a spherical component e" and a strain deviator e'.
e" = 'A±^l±Jl (248)
e'l = ei — e'
4 = 62  e" = "^ ^ ^* (249)
e'g = 63  e" =
Hooke's law may then be written in terms of the stress and strain devi
ators as follows:
a\ = 2Ge[ (250)
There are two analogous equations for the other two principal stresses
and strains. Hooke's law in terms of the spherical components is given
by
o"" + <^2 + 03' = 3K(e{' f 62' + 63') (251)
where k is the volumetric modulus of elasticity.
When the stress deviator is referred to three arbitrary orthogonal axes
X, y, z, the principal components may be found in a manner analogous to
the method for determining the principal stresses of an arbitrary state of
stress. The principal stress deviators are the roots of the cubic equation
(a'r  J, a'  J3 = (252)
The coefficients J 2 and J 3 are invariants of the stress deviator, i.e., their
values are independent of the coordinate system in which the stress devi
ator is expressed. Expre.s.sions for J2 and J3 are found below. These
quantities are frequently useful in the mathematical theory of plasticity.
W,y + w.y + w,y]
Hiiar  a^y + (a,  a,y + (^3  a,y] (253)
= —l/r.
J, =
<^x Txy T
^yX <^ y Tyz
' zy
0,
= Vd{<y\y + W,y + {a'.y]
= K7[(2oi — 02 — 03) (2(72 — 03 — 0"i)(2cr3  ax — 02)] (254)
52 Mechanical Fundamentals [Chap. 2
21 5. Strain Energy
The elastic strain energy is the amount of energy expended by the action
of external forces in deforming an elastic body. Essentially all the work
performed during elastic deformation is stored as elastic energy, and this
energy is recovered on the release of the load. Energy (or work) is equal
to a force times the distance over which it acts. In the deformation of
an elastic body the force and deformation increase linearly from initial
values of zero, so that the average energy is equal to onehalf of their
product. This is also equal to the area under the loaddeformation curve.
U = yiPb (255)
For an elemental cube that is subjected to only a tensile stress along the
sc axis, the elastic strain energy is given by
U = KP5 = ma,dA){e,dx)
= y2{(T^e,){dA dx) (256)
Equation (256) describes the total elastic energy absorbed by the ele
ment. Since dA dx is the volume of the element, the strain energy 'per
unit volume, Uq, is given by
Uo = y2CTA = l^ = Vze.'E (257)
Note that the lateral strains which accompany deformation in simple
tension do not enter in the expression for strain energy because no forces
exist in the directions of the strains.
By the same reasoning, the strain energy per unit volume of an ele
ment subjected to pure shear is given by
Uo = V2T.yy.y = l%= V^llyG (258)
The relationship for pure uniaxial deformation and pure shear can be
combined by the principle of superposition to give the elastic strain
energy for a general threedimensional stress distribution.
f/o = y2{<yxex + Oyey + a^e^ f Txylxy + r^^^xz + ry^iy^ (259)
Substituting the equations of Hooke's law [Eqs. (223) and (224)] for
the strains in the above expression results in an expression for strain
Sec. 21 5] Stress and Strain Relationships for Elastic Behavior 53
energy per unit volume expressed solely in terms of the stress and the
elastic constants.
+ ^ (jI + tI + tD (260)
Also, by substituting Eqs. (231) into Eq. (259) the stresses are elimi
nated, and the strain energy is expressed in terms of strains and the
elastic constants.
C/o = V2\^' + G{e.' + e,' + e;~) + ^Giyl + tL + ^D (261)
It is interesting to note that the derivative of Uq with respect to
any strain component gives the corresponding stress component. Thus,
dUa/de^ = XA + 2Gej, = a^ [compare with Eqs. (231)].
BIBLIOGRAPHY
Love, A. E. H.: "A Treatise on the Mathematical Theory of Elasticity," 4th ed.,
Dover Publications, New York, 1949.
Southwell, R. v.: "An Introduction to the Theory of Elasticity," 2d ed., Oxford
University Press, New York, 1941.
Timoshenko, S. P., and J. N. Goodier: "Theory of Elasticity," 2d ed., McGrawHill
Book Company, Inc., New York, 1951.
Wang, C. T.: "Applied Elasticity," McGrawHill Book Company, Inc., New York,
1953.
Chapter 3
ELEMENTS OF
THE THEORY OF PLASTICITY
31 . Introduction
The theory of plasticity deals with the behavior of materials in the
region of strain beyond which Hooke's law is no longer valid. The
mathematical description of the plastic deformation of metals is not
nearly so well developed as the description of elastic deformation by
means of the theory of elasticity because plastic deformation is much
more complicated than elastic deformation. For example, in the plastic
region of strain, there is no simple relationship between stress and strain
as there is for elastic deformation. Moreover, elastic deformation
depends only on the initial and final states of stress and is independent
of the loading path, but for plastic deformation the plastic strain depends
not only on the final load but also on the path by which it was reached.
The theory of plasticity is concerned with a number of different types
of problems. From the viewpoint of design, plasticity is concerned with
predicting the maximum load which can be applied to a body without
causing excessive yielding. The yield criterion^ must be expressed in
terms of stress in such a way that it is valid for all states of stress. The
designer is also concerned with plastic deformation in problems where
the body is purposely stressed beyond the yield stress into the plastic
region. For example, plasticity must be considered in designing for
processes such as autofrettage, shrink fitting, and the overspeeding of
rotor disks. The consideration of small plastic strains allows economies
in building construction through the use of the theory of limit design.
The analysis of large plastic strains is required in the mathematical
treatment of the plastic forming of metals. This aspect of plasticity
1 The determination of the Hmiting load between elastic and plastic behavior is also
generally covered in strength of materials. However, because it is necessary to adopt
a yield criterion in the theories of plasticity, this topic is covered in the chapter on
plasticity.
54
Sec. 32J
Elements oF the Theory of Plasticity 55
will be considered in Part Four. It is very difficult to describe, in a
rigorous analytical way, the behavior of a metal under these conditions.
Therefore, certain simplifying assumptions are usually necessary to obtain
a tractable mathematical solution.
Another aspect of plasticity is concerned with acquiring a better under
standing of the mechanism of the plastic deformation of metals. Interest
in this field is centered on the imperfections in crystalline solids. The
effect of metallurgical variables, crystal structure, and lattice imperfec
tions on the deformation behavior are of chief concern. This aspect of
plasticity is considered in Part Two.
32. The Flow Curve
The stressstrain curve obtained by uniaxial loading, as in the ordinary
tension test, is of fundamental interest in plasticity when the curve is
plotted in terms of true stress a and true strain e. True stress is given
by the load divided by the instantaneous crosssectional area of the speci
men. True strain is discussed in the next section. The purpose of this
section' is to describe typical stressstrain curves for real metals and to
compare them with the theoretical flow curves for ideal materials.
The true stressstrain curve for a typical ductile metal, such as alumi
num, is illustrated in Fig. 3la. Hooke's law is followed up to some yield
[a] [b)
Fig. 31 . Typical true stressstrain curves for a ductile metal.
stress oQ. (The value of oq will depend upon the accuracy with which
strain is measured.) Beyond cro, the metal deforms plastically. Most
metals strainharden in this region, so that increases in strain require
higher values of stress than the initial yield stress cro. However, unlike
the situation in the elastic region, the stress and strain are not related by
' See Chap. 9 for a more complete discussion of the mathematics of the true stress
strain curve.
56 Mechanical Fundamentals [Chap. 3
any simple constant of proportionality. If the metal is strained to point
A, when the load is released the total strain will immediately decrease
from €i to C2 by an amount a/E. The strain decrease ei — eo is the
recoverable elastic strain. However, the strain remaining is not all perma
nent plastic strain. Depending upon the metal and the temperature, a
small amount of the plastic strain €2 — €3 will disappear with time. This
is known as anelastic behavior. ^ Generally the anelastic strain is neg
lected in mathematical theories of plasticity.
Generally the stressstrain curve on unloading from a plastic strain
will not be exactly linear and parallel to the elastic portion of the curve
(Fig. 316). Moreover, on reloading the curve will generally bend over
as the stress approaches the original value of stress from which it was
unloaded. With a little additional plastic strain, the stressstrain curve
becomes a continuation of what it would have been had no unloading
taken place. The hysteresis behavior resulting from unloading and load
ing from a plastic strain is generally neglected in plasticity theories.
A true stressstrain curve is frequently called a flow curve because it
gives the stress required to cause the metal to flow plastically to any
given strain. Many attempts have been made to fit mathematical equa
tions to this curve. The most common is a power expression of the form
a = Ke (31)
where K is the stress at e = 1.0 and n, the strainhardening coefficient,
is the slope of a loglog plot of Eq. (31). This equation can be valid
only from the beginning of plastic flow to the maximum load at which
the specimen begins to neck down.
Even the simple mathematical expression for the flow curve that is
given by Eq. (31) can result in considerable mathematical complexity
when it is used with the equations of the theory of plasticity. Therefore,
in this field it is common practice to devise idealized flow curves which
simplify the mathematics without deviating too far from physical reality.
Figure 32a shows the flow curve for a rigid, perfectly plastic material.
For this idealized material, a tensile specimen is completely rigid (zero
elastic strain) until the axial stress equals cro, whereupon the material
flows plastically at a constant flow stress (zero strain hardening). This
type of behavior is approached by a ductile metal which is in a highly
coldworked condition. Figure 326 illustrates the flow curve for a
perfectly plastic material with an elastic region. This behavior is
approached by a material such as plaincarbon steel which has a pro
nounced yieldpoint elongation (see Sec. 55). A more realistic approach
is to approximate the flow curve by two straight lines corresponding to
1 Anelasticity is discussed in greater detail in Chap. 8.
Sec. 33]
Elements of the Theory of Plasticity
57
Fig. 32. Idealized flow curves, (a) Rigid ideal plastic material; (b) ideal plastic
material with elastic region; (c) piecewise linear (strainhardening) material.
the elastic and plastic regions (Fig. 32c). This type of curve results in
somewhat more complicated mathematics.
33. True Strain
Equation (11) describes the conventional concept of unit linear strain,
namely, the change in length referred to the original unit length.
e = y = ^ / dL
Lid Lio Jlo
This definition of strain is satisfactory for elastic strains where AL is very
small. However, in plastic deformation the strains are frequently large,
and during the extension the gage length changes considerably. Ludwik^
first proposed the definition of true strain, or natural strain, e, which obvi
ates this difficulty. In this definition of strain the change in length is
referred to the instantaneous gage length, rather than to the original
gage length.
■l^
~ Lo L^
or
^dL , L
^ = In J
u
Li Ls — L2 .
L2
(32)
(33)
The relationship between true strain and conventional linear strain
follows from Eq. (11).
e+l =
AL
Lo
L
L — Lo _ L
Lo Lo
e = In ^ = In (e 1 1)
(34)
* P. Ludwik, "Elemente der technologischen Mechanik," SpringerVerlag OHG.
Berlin, 1909.
58 Mechanical Fundamentals [Chap. 3
The two measurements of strain give nearly identical results up to
strains of about 0.1.
Because the volume remains essentially constant during plastic defor
mation, Eq. (33) can be written in terms of either length or area.
6 = In ^ = In 4^ (35)
Also, because of constancy of volume, the summation of the three princi
pal strains is equal to zero.
61 + 62 + €3 = (36)
This relationship is not valid for the principal conventional strains.
The advantage of using true strain should be apparent from the follow
ing example: Consider a uniform cylinder which is extended to twice its
original length. The linear strain is then e = (2Lo — Lo)/Lo = 1.0, or a
strain of 100 per cent. To achieve the same amount of negative linear
strain in compression, the cylinder would have to be squeezed to zero
thickness. However, intuitively we should expect that the strain pro
duced in compressing a cylinder to half its original length would be the
same as, although opposite in sign to, the strain produced by extending
the cylinder to twice its length. If true strain is used, equivalence is
obtained for the two cases. For extension to twice the original length,
e = In (2Lo/Lo) = In 2. For compression to half the original length,
€ = In [(Lo/2)/Lo] = In M =  In 2.
34. yielding Criteria for Ductile Metals
The problem of deducing mathematical relationships for predicting
the conditions at which plastic yielding begins when a material is sub
jected to a complex state of stress is an important consideration in the
field of plasticity. In uniaxial loading, plastic flow begins at the yield
stress, and it is to be expected that yielding under a situation of com
bined stresses is related to some particular combination of the princi
pal stresses. A yield criterion can be expressed in the general form
F(ai,a2,(T3,ki, . . .) = 0, but there is at present no theoretical way of cal
culating the relationship between the stress components to correlate yield
ing in a threedimensional state of stress with yielding in the uniaxial
tension test. The yielding criteria are therefore essentially empirical
relationships. At present, there are two generally accepted theories for
predicting the onset of yielding in ductile metals.
Maximumshearstress Theory
The maximumshearstress theory, sometimes called the Tresca, Cou
lomb, or Guest yield criterion, states that yielding will occur when the
(7i — 03
= ro= 2
or 01 — 03 = (To
This is sometimes written as
Sec. 34] Elements of the Theory of Plasticity 59
maximum shear stress reaches a critical value equal to the shearing yield
stress in a uniaxial tension test. From Eq. (215), the maximum shear
stress is given by ^
_ (Ti — 03
Tmax Q \0~l )
where en is the algebraically largest and as is the algebraically smallest
principal stress.
For uniaxial tension ai = ao, ao = (Ts = 0, where ao is the yield strength
in simple tension. Therefore, the shearing yield stress for simple tension
To is equal to onehalf of the tensile yield stress.
O"0
To — ^ r;
Substituting these values into the equation for the maximum shear stress
results in
(38)
(39)
(Ti  <Js = a[  a, = 2k (310)
where a^ and a'^ are the deviators of the principal stresses and k is the
yield stress for pure shear, i.e., the stress at which yielding occurs in
torsion, where ai = —as.
The maximumshearstress theory is in good agreement with experi
mental results, being slightly on the safe side, and is widely used by
designers for ductile metals. It has replaced the older and far less accu
rate maximumstress theory, Rankine's theory.
Prager and Hodge ^ have pointed out that in certain plasticity problems
the simple relations of Eq. (39) or (310) cannot be used as the yielding
conditions since it is not known which of the three principal stresses is the
largest. In this case, the much more complicated general form of the
equation, which is given below, must be used.
4/2'  27J3'  36^2/2^ + 96/CV2  64A;« = (311)
J 2 and Js are the invariants of the stress deviator (see Sec. 214). Obvi
ously, such a complex relation will result in very cumbersome mathe
matics. It is for this reason that the yielding criterion that is discussed
next is preferred in most theoretical work.
^ W. Prager and P. G. Hodge, Jr., "Theory of Perfectly Plastic Solids," p. 23,
John Wiley & Sons, Inc., New York. 1951.
60 Mechanical Fundamentals [Chap. 3
Von Mises, or Distortionenergy, Theory
A somewhat better fit with experimental results is provided by the
yield criterion given in Eq. (312).
(70 = ^ [((Ti  a,y + ((72  a^y + ((73  (7i)2]^^ (312)
V2
According to this criterion, yielding will occur when the differences
between the principal stresses expressed by the righthand side of the
equation exceed the yield stress in uniaxial tension, ao. The develop
ment of this yield criterion is associated with the names of Von Mises,
Hencky, Maxwell, and Huber. Von Mises proposed this criterion in the
invariant form given by Eq. (313) primarily because it was mathemati
cally simpler than the invariant form of the maximumshearstress theory
given by Eq. (311). Subsequent experiments showed that Eq. (313)
provides better overall agreement with combined stressyielding data
than the maximumshearstress theory.
^2  fc2 = (313)
J2 is the second invariant of the stress deviator, and k is the yield stress
in pure shear.
A number of attempts have been made to provide physical meaning
to the Von Mises yield criterion. One commonly accepted concept is
that this yield criterion expresses the strain energy of distortion. On the
basis of the distortionenergy concept, yielding will occur when the strain
energy of distortion per unit volume exceeds the strain energy of distor
tion per unit volume for a specimen strained to the yield stress in uniaxial
tension or compression. The derivation of Eq. (312) on the basis of
distortion energy is given below. Another common physical interpreta
tion of Eq. (312) is that it represents the critical value of the octahedral
shear stress (see Sec. 37).
The total elastic strain energy per unit volume (see Sec. 215) can be
divided into two components, the strain energy of distortion, Uq, and the
strain energy of volume change, Uq. To illustrate the resolution of total
strain energy into its components, consider Fig. 33. This figure illus
trates the point established in Sec. 214 that a general threedimensional
state of stress can be expressed in terms of a spherical or hydrostatic
component of stress, a", and a stress deviator, a'. Because experiments
have shown 1 that up to rather large values of hydrostatic pressure a
hydrostatic state of stress has no effect on yielding, it is valid to assume
that only the stress deviator can produce distortion. Therefore, the
1 P. W. Bridgman, "Studies in Large Plastic Flow and Fracture," McGrawHill
Book Company, Inc., New York, 1952,
Sec. 34] Elements of the Theory of Plasticity 61
strain energy of distortion will be based on the stress deviator. It repre
sents only the strain energy associated with changing the shape of the
specimen and neglects the strain energy associated with changes in
volume.
The strain energy of distortion will be determined by first calculating
the strain energy of volume change and then subtracting this term from
Oz Og 02
Fig. 33. Resolution of stress into hydrostatic stress and stress deviator.
the total strain energy. Referring again to Fig. 33, the strain energy
per unit volume associated with a volume change will be
jrii ■, / n n I i / It n 1 , / 11 ir
Uo = /'201 ei + >^02 62 + >203 ^3
Referring to the definitions for the deviator component of strain given in
Sec. 214 and letting o„i equal the hydrostatic component of stress, or
average stress,
Uo = H<T.(ei + 62 + 63) = V2<rmA (314)
However, from Eq. (229) A = (Xm/n, so that Eq. (314) becomes
U'o'^l"^ (315)
Since U'o = Uo — U[' , the strain energy of distortion can be determined
by using Eq. (260) for the total strain energy C/o.
1 12
f/o = 2^ (O"!^ + 02 1 03^) — p (olO2 + 02(r3 "I" Oldi) " 9 "^ (316)
However, since (Xm = (oi f 02 f (Ts)/3 and k = E'/[3(l — 2v)], Eq.
(316) reduces to
C/'O = —^ [(<^l  ^2)2 + (<T2  a,r + (<T3  CT.y] (317)
For a uniaxial state of stress, <ti — tro, cr2 = 03 = 0.
C/'o = ^'^o^ (318)
62 Mechanical Fundamentals [Chap. 3
Therefore, the distortionenergy yield criterion can be written
or <T0 = ^ [{<Jl  <T2y + (<r2  a,y + ((73  (7l)2]^^ (319)
For a condition of pure shear, such as occurs in torsion, t = a.
O"! ^= fO Cr2 = Cj ^ — (To
Therefore, the strain energy of distortion for this state of stress is given by
Uo=^^<^'=^^r^ (320)
If for any type of stress system yielding begins when the strain energy of
distortion reaches a critical value, the ratio between this critical value
for uniaxial stress and pure shear can be obtained by equating Eqs.
(318) and (320).
1 + V 2 1 + '^ 2
TO = ^(70 = 0.577(70 (321)
V3
Thus, if the distortionenergy theory is a valid yielding criterion, the
yield strength in shear, as determined from a torsion test, should be
0.577 times the tensile yield strength. Actual data show that the shear
yield stress falls between 0.5 and 0.6 of the tensile yield stress, with the
average occurring close to the predicted value. It should be noted that
the maximumshearstress theory predicts that tq = 0.50(7o. The better
agreement shown by the distortionenergy theory for these two different
types of tests is one reason for preferring the distortionenergy theory of
yielding.
35. Combined Stress Tests
The conditions for yielding under states of stress other than uniaxial
and torsion loading can be conveniently studied with thinwall tubes.
Axial tension can be combined with torsion to give various combinations
of shear stiess to normal stress intermediate between the values obtained
separately in tension and torsion. For the combined axial tension and
(T2 =
(322)
Sec. 35] Elements oF the Theory of Plasticity 63
torsion, the principal stresses from Eq. (27) are
^1 = 2" + (^T + ^j
2 V 4 """7
Therefore, the maximumshearstress criterion of yielding is given by
fey +^ fey =1 ^'''^
and the distortionenergy theory of yielding is expressed by
Both equations define an ellipse. Figure 34 shows that the experi
mental results 1 agree best with the distortionenergy theory.
0.6
0.5
0.4
I?
^ 0.3
0.2
0.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fig. 34. Comparison between maximumshearstress theory and dist9rtionenergy
(Von Mises) theory.
Another type of combined stress test is to subject thinwall tubes to
axial load and internal hydrostatic pressure.^ Since the stress in the
radial direction is negligible (03 = at the outer free surface), this test
provides a biaxial state of stress.
For the planestress condition, the distortionenergy theory of yielding
can be expressed mathematically by
(325)
1
>
"<
 Distort 10
n ene
rgy
~
^^
>v
Mox/r
num s
hear
strest
^
^
\,
\^
\
\\
^
Ol + Oo
aiCfy
ao^
1 G. I. Taylor and H. Quinney, Proc. Roy. Soc. (London), vol. 230A, pp. 323362,
1931.
2 W. Lode, Z. Phijsik, vol. 3G, pp. 913939, 1926.
64
Mechanical Fundamentals
[Chap. 3
Distortion
energy theory
This is the equation of an ellipse whose major semiaxis is x/^ co and
whose minor semiaxis is ^/% oq.
A convenient way of comparing yielding criteria for a twodimensional
state of stress is with a plot such as Fig. 35. Note that the maximum
shearstress theory and the distor
tionenerg3'^ theory predict the same
yield stress for conditions of uni
axial stress and balanced biaxial
stress (<Ti = a 2). The greatest di
vergence between the two theories
occurs for a state of pure shear
((71 = —02). It has already been
shown that for this state of stress
the shearstress law predicts a yield
stress which is 15 per cent lower
than the value given by the distor
tionenergy criterion.
A very sensitive method of dif
ferentiating between the two yield
criteria is the procedure adopted by
Lode of determining the effect of
the intermediate principal stress on
yielding. According to the maximumshearstress law, there should be no
effect of the value of the intermediate stress 02. Thus, (oi — 03) /oq = 1.
For the distortionenergy theory, to account for the influence of the
intermediate principal stress, Lode introduced the parameter n, called
Lode's stress parameter.
Maximum shear
stress theory
Fig. 35. Comparison of yield criteria for
plane stress.
M =
2(T2 — g"3 — 0"]
O"! — 03
(326)
Solving this equation for 02 and eliminating 02 from Eq. (312) results in
(7 1 — 03 2
<T0 (3 + M^)
2\H
(327)
Experimental data plot much better against Eq. (327) than against the
maximumshearstress equation, indicating that the intermediate princi
pal stress has an influence on yielding.
Another contribution of Lode was the introduction of a strain
parameter v.
V = 2 A62  A63  Aei .3_2g)
Aei — Aes
where Ae is a finite increment of strain. A plot of n against v should
Sec. 36] Elements of the Theory of Plasticity 65
yield a straight line at 45° to the axis if the metal behaves according to
the LevyVon Mises equations of plasticity (see Sec. 39). Most metals
show some slight but systematic deviation from Lode's relationship n = v.
36. Octahedral Shear Stress and Shear Strain
The octahedral stresses are a particular set of stress functions which
are important in the theory of plasticity. They are the stresses acting
on the faces of a threedimensional octahedron which has the geometric
property that the faces of the planes make equal angles with each of the
three principal directions of stress. For such a geometric body, the angle
between the normal to one of the faces and the nearest principal axis is
54°44', and the cosine of this angle is l/\/3.
The stress acting on each face of the octahedron can be resolved^ into
a normal octahedral stress oQct and an octahedral shear stress lying in the
octahedral plane, Xoct. The normal octahedral stress is equal to the hydro
static component of the total stress.
Coot = "^ + "3^ + "^ = a" (329)
The octahedral shear stress r„ct is given by
root = M[(^l  <J,Y + (cr2  (TzY + (C73  (Ti)']^^ (330)
Since the normal octahedral stress is a hydrostatic stress, it cannot pro
duce yielding in solid materials. Therefore, the octahedral shear stress
is the component of stress responsible for plastic deformation. In this
respect, it is analogous to the stress de viator.
If it is assumed that a critical octahedral shear stress determines yield
ing, the failure criterion can be written as
1 ^, ^2
Toot = r, [(o"l — 02)^ + (02 — 03)2 + ((T2 — Ol)]'^ = —^ (To
or oo = ~ [(^1  ^^y + (<^2  (T^y f ((73  a.y^'' (331)
Since Eq. (331) is identical with the equation already derived for the
distortionenergy theory, the two yielding theories give the same results.
In a sense, the octahedral theory can be considered the stress equivalent
of the distortionenergy theory. According to this theory, the octahedral
1 A. Nadai, "Theory of Flow and Fracture of Solids," 2d ed., vol. I, pp. 99105,
McGrawHill Book Company, Inc., New York, 1950.
66 Mechanical Fundamentals [Chap. 3
shear stress corresponding to yielding in uniaxial stress is given by
\/2
Toct= ^(70  0.471(70 (332)
Octahedral strains are referred to the same threedimensional octahe
dron as the octahedral stresses. The octahedral linear strain is given by
€oct = o (333)
Octahedral shear strain is given by
Toot = %[(^l  ^2)' + (62  63)2 + (63  61)^]^^ (334)
37. Invariants of Stress and Strain
It is frequently useful to simplify the representation of a complex state
of stress or strain by means of invariant functions of stress and strain.
If the plastic stressstrain curve (the flow curve) is plotted in terms of
invariants of stress and strain, approximately the same curve will be
obtained regardless of the state of stress. For example, the flow curves
obtained in a uniaxialtension test and a biaxialtorsion test of a thin tube
with internal pressure will coincide when the curves are plotted in terms
of invariant stress and strain functions.
Nadai^ has shown that the octahedral shear stress and shear strain are
invariant functions which describe the flow curve independent of the type
of test. Other frequently used invariant functions are the effective, or
significant, stress and strain. These quantities are defined by the follow
ing equations for the case where the coordinate axes correspond to the
principal directions:
Effective or significant stress
^ = ^ [(<ri  <T2y + ((T2  <x,y + ((73  a{}T (335)
Effective or significant strain
^ = ^ [(ei  62)2 + (62  63)2 + (63  61)^]^^ (336)
Note that both effective stress and effective strain reduce to the axial
normal component of stress and strain for a tensile test. These values
1 A. Nadai, J. Appl. Phys., vol. 8, p. 205, 1937.
)ec.
381 Elements oF the I heory of Plasticity 67
are also related to the octahedral shearing stress and strain, as can be
seen by comparing Eqs. (330) and (334) with the above equations.
\/2 /
Toct = — o— 0 Toct = V 2 e (337)
Drucker^ has pointed out that there are a large number of different
functions of stress and strain which might serve as invariant stress and
strain parameters. For example, he shows that combined stress data for
aluminumalloy tubes show better agreement when the eciuivalent shear
ing stress Teq, defined below, is plotted against octahedral shearing strain
instead of Xoct being plotted against Toct
— Toct
0^:)'
(338)
where J 2 and J3 are invariants of the stress deviator. There appears to
be no theoretical or experimental justification for choosing invariant stress
and strain parameters other than the closeness of agreement with the
data and mathematical convenience.
38. Basis of the Theories of Plasticity
The development of a generalized theory of plasticity, wnth the same
wide applicability as the theory of elasticity, has not progressed rapidly
because of the complexity of the problem. The inherent difficulty in
developing a simple mathematical description of plasticity lies in the fact
that plastic deformation is essentially an irreversible process. While
elastic deformation depends only on the initial and final states of stress
or strain and therefore the results are independent of the path along
which the load is reached, in plastic deformation the total plastic strain
depends not only on the final load but also on the path by which it was
reached. Therefore, in plastic deformation, the type of load cycle deter
mines the strain increment. The final value of a plasticstrain component
is given by the integral of the increments of the plasticstrain component
over the loading history that the material has undergone.
A particular condition of loading which simplifies the analysis is 'propor
tional loading. For proportional loading, the stress components increase
in constant ratio to each other.
dcFl _ da 2 _ d(Tz /q OQN
(T\ 0'2 O'S
For this type of loading, the strains can be expressed in terms of the final
stress state because the final stress state specifies the stress history.
' D. C. Drucker, J. Appl. Mech., vol. 16, pp. 349357, 1949.
68 Mechanical Fundamentals [Chap. 3
Mathematical theories of plasticity can be divided roughly into two
types. Deformation theories relate the stress to the strain, while flow
theories relate the stress to the strain rate, or the velocity of strain.
Deformation theories utilize an averaging process over the entire defor
mation history and relate the total plastic strain to the final stress. This
tj^pe of theory is valid when the material is subjected to proportional
loading, but it is not generally considered to be reliable^ when the direc
tion of loading is changed during the test. Flow theories consider a suc
cession of infinitesimal increments of distortion in which the instantaneous
stress is related to the increment of the strain rate. Because a flow theory
considers the instantaneous deformation, it is better able to describe large
plastic deformations.
There are a number of general assumptions which are common to all
plasticity theories. The metal is considered to be continuous and iso
tropic. The principal axes of plastic stress and strain are assumed to
coincide at all times. Time effects are usually neglected, so that visco
elastic materials are excluded from the theories presented in this chapter.
For the values of stress usually encountered, it is a very good assumption
to consider that volume remains constant. This also leads to the useful
relation that the sum of the principal true strains is equal to zero.
61 + €2 + 63 =
Constancy of volume also requires that Poisson's ratio must increase from
its elastic value to a value of 0.5 for the plastic condition. Experiments
show that Poisson's ratio increases with progressive plastic strain to this
limiting value, but frequently the incorrect assumption is made that
V = 0.5 for all values of plastic strain.
Unfortunately, there is no simple relationship between stress and strain
in the plastic region such as exists with elastic deformation. An obvious
simplification is to assume that plastic flow occurs at a constant value of
flow stress, i.e., that there is no strain hardening. Plasticity theory
based on ideal plastic behavior has been developed further than theories
which consider the strain hardening of the metal. One way to take strain
hardening into consideration is to use experimental data plotted as invari
ant stressstrain functions. In analyses of forming operations involving
large plastic strains, it is common practice to allow for strain hardening
by using an average value of yield stress.
The formation of many plasticity theories is based on the premise
that the stress deviator is proportional to the strain increment. This is
equivalent to saying that Lode's stress and strain parameters are equal,
^ Arguments to show that deformation theories of plasticity should be valid for
loading paths other than proportional loading have been given by B. Budiansky,
J. Appl. Mech., vol. 26, no. 2, pp. 259264, 1959.
Sec. 39] Elements of the Theory of Plasticity 69
n = V. Although deviations from Lode's relationship have been shown
by experiment, it appears that the proportionality between stress devi
ator and strain increment is a reasonably good approximation.
To provide additional simplification to the analysis, it is often assumed
that the body acts as a rigid plastic material. With this assumption,
all elastic strain is neglected, and the total strain is considered to be
entirely plastic. This is a suitable assumption when the plastic strain is
large, so that the elastic strains are negligible by comparison. However,
in many problems the body is strained only slightly beyond the yield
stress, so that the elastic and plastic strains are of comparable magnitude.
For this situation, it is necessary that the elastic strains be considered in
the analysis. The total strain is then the sum of the elastic and plastic
strain.
^ij = 4 + 4 (340)
However, because of the assumption of constancy of volume, the plastic
component of the hydrostatic component of strain must be equal to zero.
e^" = H(^/ + 62^ + 63^) =
Therefore, the plasticstrain deviator is equal to the plastic strain.
e/' = er €2^ = er e,^ = e^ (341)
39. Flow Theories
Rigid Ideal Plastic Material
A flow theory for a rigid ideal plastic material based on the propor
tionality between stress deviator and strain rate is the outgrowth of work
by St. Venant, Levy, and Von Mises. The Levy Von Mises equations
are given below for a general coordinate system. X is a proportionality
constant, and <rm is the hydrostatic component of stress. Note that a
dot over the symbol for strain indicates the time derivative of strain, i.e.,
the strain rate.
Cx ~ CFm = fx — 2Xex Txy = ^^jxy
a'y = 2Xey Ty^ = Xjy, (342)
a^ = 2X63 Txz = Xjxz
In terms of the principal stresses, the Levy Von Mises equations can
be written
a^ = 2Xei ffo = 2X€2 a"3 — 2Xe3 (343)
These equations are similar to the equations of viscosity for an incom
pressible fluid. The important difference is that for the case of the fluid
the proportionality constant X is a true material constant, the coefficient
70 Mechanical Fundamentals [Chap. 3
of viscosity. For the case of the plastic body, the value of X depends on
the vakies of stress and strain. X can be evaluated when the yield cri
terion is established.
The Von Mises yield criterion is given by
or 2J, = {a[Y + {a',Y + {c',Y^ ^' (344)
Substituting Eqs. (343) into Eq. (344) results in
X2(6i2 + e^^ + h') = ^' (345)
The quantity ei^ + €2^ + €3^ is an invariant of strain rate. Substituting
Eq. (345) back into Eqs. (343) gives
> = V^ opei (o_Aa\
' [3(6i2 + e^^ H k,^W ^ ^
Completely analogous equations follow for a'^ and 03.
Equations (343) can be written
Z<Ti — 02 — 03 = 37 dei
at
2o2 ~ <^i ~ <^3 = "^ '^^2 (347)
Z(T3 — (Ti — (T2 — —r; Ci^3
at
Eliminating Q\/dt from these equations results in
2ai — (T2 — (7z _ dei
2a2 — cs — o"! de2
2cri — 02 — Cs _ dei
2(73 — Ci — (72 C?€3
(348)
The above two equations, plus the constancyofvolume relationship
ei + €2 + €3 = 0, constitute a system of differential equations that must
be integrated over a particular stress path or strain path for the solution
of a particular problem.
ElasticPlastic Material
The extension of the LevyVon Mises equations to the consideration
of both elastic and plastic strains has been primarily the work of Prandtl
and Reuss. In discussing this theory it will be necessary to differentiate
between the elastic strain e^ and the plastic strain e^. Assuming that
Sgc. 39] Elements oF the Theory of Plasticity 71
the rate of change of plastic strain is proportional to the stress deviator
results in
2^6/' = \a[ 2^62^' = \<j', 2Gi,P' = \a', (349)
The time derivative of Hooke's law expressed in terms of stress and strain
deviators [Eq. (250)] gives the corresponding equations for elastic strain.
2G'6i^' = &[ 2G'62^' = &2 2(?e3^' = &'s (350)
Combining Eqs. (349) and (350) results in expressions for the time
derivative of total strain.
2G€[ = (Ti + XcTi 2G'e2 = 0^2 + Xo2 26^63 = ^3 "f \a^ (351)
If it is assumed that the Von Mises criterion of yielding applies and that
there is no strain hardening,
J2 = F J2 =
From Eq. (344)
J2 = a[{a[r + a',{a',r f a[(a',r = (352)
This expression can be used to eliminate the proportionality constant X
from Eq. (351). However, to simplify the algebra, the quantity Uq is
introduced.' U'o is the rate of change of strain energy involved in dis
tortion, as opposed to the strain energy required for volume change.
U'o = o'ifi + aU'^ + cr'^e's (353)
By using Eqs. (352) and (353) and the yield criterion J2 = fc^ it is
possible to obtain the relationship
2GUo = 2XA;2 (354)
The stressstrain relations of Reuss's equations are obtained by substi
tuting Eq. (354) into Eq. (352) and solving for the stress rate.
a', = SG'f^l^^;) (355)
= 2G
(., 3L^'o A
These equations give the rate of change of the stress deviator as long as
J 2 = k"^ and £/[, > 0. To get the rate of change of stress, it is necessary
to remember that ai = a[ f a^ . From Eq. (251)
a" = 3/ce^" (356)
^ This derivation follows the procedure given by Prager and Hodge, op. cit., pp.
2729.
72 Mechanical Fundamentals
[Ch
ap.
When the stress is in the elastic range, or in unloading from the plastic
region, Eqs. (355) do not apply. The proper equations are given by the
elasticity equations like Eq. (250).
310. Deformation Theories
Hencky proposed that for small strains the stress deviator could be
considered proportional to the strain deviator.
(J = 2Gp^'
(357)
Elastic strains are neglected in Eq. (357). Gp is a plastic shear modulus
which varies depending upon the values of stress and strain. Because
of the assumption of constancy of volume, e" = 0, and e = e. There
fore, Eq. (357) can be expanded in terms of principal stresses and strains
to give
ei
€2 =
es
2(Ti — a2 — ^3 _ 1
2a'2 — Ci — CTs 1
QGp ^ 3Gp
2cr3 — (Ti — ^2 _ 1
QGp ~ 3^
O"! — 2 (^^2 + 03)
os — 2 (<^i + ^2)
Ep
J_
E p
J_
Ep
Ol — 2 (^^2 + 03)
02 — 2 ^^^ "*" '^^^
03 — 2 ^^^ '^ '^^^
(358)
The analogy is apparent between the righthand side of Eqs. (358) and
the familiar equations of elasticity expressing strain in terms of the
principal stresses [Eqs. (223)].
For the plastic case, Poisson's ratio has
been taken equal to 3^. Ep can be
considered to be a plastic modulus that
is actually a variable depending upon
the stress and strain. The evaluation
of Ep from an invariant stressstrain
curve is shown in Fig. 36.
J_ ^ e
Ep ff
(359)
Significant strain
Fig. 36. Definition of Ep.
Nadai^ has developed relationships
similar to Eqs. (358) based on the equal
ity of Lode's stress and strain param
eters. The fact that 11 = v leads to the conclusion that the ratios of
the principal shearing stresses to the principal shearing strains are equal.
1 A. Nadai, "Plasticity," pp. 7779, McGrawHill Book Company, Inc., New York,
1931.
)ec.
311]
Elements of the Theory of Plasticity 73
and from these three relationships the equations can be derived. It is
for this reason that relationships like Eqs. (358) are sometimes called
Nadai's equations.
In a deformation theory, such as is given by the Hencky or Nadai
equations, the total plastic strain is proportional to the stress de viator,
while in a flow theory, such as is given by the Reuss equations, the
increments of plastic strain are proportional to the stress deviator. The
Hencky theory gives results in agreement with the incremental or flow
theory provided that the principal axes of stress and strain remain coinci
dent during the straining process and provided that proportional loading
is maintained. The Hencky theory is not satisfactory for large defor
mations, but it is often used for small plastic strains because it offers
certain mathematical conveniences.
311. Twodimensional Plastic Flow — Plane Strain
In many practical problems, such as rolling and drawing, all displace
ments can be considered to be limited to the xy plane, so that strains in
Punch
Plastic
metal
// // merai // //
<i^////////////////////////A.
Punch
f^=0
Rigid ^^■ • ^ ■ .■ • ^ ■  ■ ^v'J v Rigid
^iPlastic^^
v//////y//?7/?/////////
(a) Kb)
Fig. 37. Methods of developing plastic constraint.
the z direction can be neglected in the analysis. This is known as a con
dition of ylane strain. When a problem is too difficult for an exact three
dimensional solution, a good indication of the deformation and forces
required can often be obtained by consideration of the analogous plane
strain problem.
Since a plastic material tends to deform in all directions, to develop a
planestrain condition it is necessary to constrain the flow in one direc
tion. Constraint can be produced by an external lubricated barrier, such
as a die wall (Fig. 37 a), or it can arise from a situation where only part
of the material is deformed and the rigid material outside the plastic
region prevents the spread of deformation (Fig. 37b).
Even though the strain in one of the principal directions is equal to
74 Mechanical Fundamentals [Chap. 3
zero for plane strain, it does not follow that there is zero stress in this
direction. It can be shown ^ that for plane strain o^ = (a^ + (Ty)/2 or
(Ts = (oi + c72)/2. If this value is substituted into the expression for the
Von Mises criterion of yielding, the yield criterion for plane strain becomes
2
(Ti — a2 = — 7= Co = 1.15oo (360)
V3
The maximumshearstress criterion of yielding can be expressed by
(Ti — cTs = (To = 2k. However, with the planestrain condition defining
the value of 03, the minimum principal stress will be 02 and the shear
stress criterion should be written
CTi — (72 = CTo = 2k (361)
In Eq. (361) k is the yield stress in pure shear. However, based on the
Von Mises criterion of yielding the relationship between the tensile yield
stress and the yield stress in shear [Eq. (321)] is given by ca = s/S k.
Therefore, Eq. (360) becomes oi — 02 = 2k. Thus, for planestrain con
ditions, the two yield criteria are equivalent, and it can be considered
that twodimensional flow will occur when the shear stress reaches a
critical value of k. Equation (361) is equally valid when written in
terms of the stress deviator.
(,; = ao = 2k (362)
312. Slipfield Theory
Consider a volume element in plane strain within a plastic region of a
body. Figure 38a represents the twodimensional state of stress with
respect to arbitrary cartesian coordinates. It is possible to determine
the principal planes such that the shear stresses vanish (Fig. 386). The
principal stresses are simply functions of the spherical component of
stress, a" , and the shearing stress k. fc is a constant throughout the
plastic region if strain hardening is neglected, but a" varies from point to
point. The maximum shear stress will occur on planes 45° to the direc
tion of the principal stresses. Thus, the critical shear stress k will first
reach its value on these planes. This condition is shown in Fig. 38c,
where it is seen that the maximum shear stress occurs in two orthogonal
directions, designated a and ,8. These lines of maximum shear stress are
called slip lines. The slip lines have the property that the shear strain
is a maximum and the linear strain is zero tangent to their direction.
However, it should be carefully noted that the slip lines just referred to
' O. Hoffman and G. Sachs, "IntroductiontotheTheoryof Plasticity for Engineers, "
p. 1]8, McGrawHill Book Company, Inc., New York, 1953.
Sec. 312]
Elements of the Theory of Plasticity
75
are not the slip lines, or slip bands, observed under the microscope on
the surface of plastically deformed metal. This latter type of slip lines
will be discussed more fully in the next chapter.
o] =a +/[
ic)
Fig. 38. Twodimensional state of stress in plane strain.
By comparing Fig. 386 and c, it is seen that the principal stresses have
a direction 45° to the slip lines. The values of the principal stresses can
be determined if a" is known since
cTi = a" \ k
(72 — 0" — K
(363)
If a" is constant throughout the region, the slip lines will be straight lines.
a< =0
Fig. 39. Slipline field at free surface.
However, if the slip Unes curve by an angle 0, the following relationships
hold:
a" j 2k(^ — constant along a line
a" — 2k4) = constant along (3 line
(364)
The slip lines at a free surface must make an angle of 45° with the
surface (Fig. 39), since there can be no resultant tangential force at a
76
Mechanical Fundamentals
[Ch
ap.
free surface. Since there is no resultant normal stress at a free surface,
01 = and by Eqs. (363) 0" = —k. Therefore, 02 = —2fc, and the trans
verse principal stress is compressive with a value of 2k.
As a further example of the use of slip lines, consider the deformation
of an ideal plastic metal by a fiat punch. ^ The friction between the face
Fig. 310. Slipline field produced by indentation of a punch.
of the punch and the metal is considered to be negligible. Plastic defor
mation will first start at the corners of the punch and will result in a
slipline field such as is shown in Fig. 310. Consider the point M.
Since this is at a free surface, the normal stress is zero and a" = k. In
accordance with Eqs. (364), the equation of this slip line may be written
a" \ 2k(f> = k. There is no change in the value of a" until we reach
point A^, where the slip line deviates from a straight line. In going from
A'' to Q, the slip line turns through an angle ^ = — 7r/2 so that its equa
tion at point Q is a" — 2/c(7r/2) = k. Since no further change takes
place in (/> in going to point R, the principal stress normal to the surface
at R is
<yin = a" ^ k = (k ^ 2k l) + k
a,, = 2k(^l+ fj
or
and
^2« = 2k
If we trace out any of the other slip lines, we shall find in the same way
that the normal stress is 2k{l \ x/2). Therefore, the pressure is uni
form over the face of the punch and is equal to
(71 = 2k[l {
i)
(365)
1 D. Tabor, "The Hardness of Metals," pp. 3437, Oxford University Press, New
York, 1951.
Sec. 312] Elements of the Theory of Plasticity 77
Since k = ao/y/S,
<Ji = cr^ax = ^ ^1 + I ) = 3(ro (3_66:
Thus, the theory predicts that fullscale plastic flow, with the resulting
indentation, will occur when the stress across the face of the punch
reaches three times the yield strength in tension.
The example described above is relatively simple and represents an
overidealized situation. However, the method of slip fields, sometimes
called the Hencky plasticsection method, is an important analytical tool
for attacking difficult problems in plasticity. It has been used in the
analysis of twodimensional problems such as the yielding of a notched
tensile bar^ and the hot rolling of a slab.^ Prager^ and Thomsen* have
given general procedures for constructing slipline fields. However, there
is no easy method of checking the validity of a solution. Partial experi
mental verification of theoretically determined slipline fields has been
obtained for mild steel by etching techniques^ which delineate the plas
tically deformed regions.
BIBLIOGRAPHY
Hill, R.: "The Mathematical Theory of Plasticity," Oxford University Press, New
York, 1950.
Hoffman, O., and G. Sachs: "Introduction to the Theory of Plasticity for Engineers,"
McGrawHill Book Company, Inc., New York, 1953.
Nadai, A.: "Theory of Flow and Fracture of Solids," 2d ed., vol. I, McGrawHill
Book Company, Inc., New York, 1950.
Phillips, A.: "Introduction to Plasticity," The Ronald Press Company, New York,
1956.
Prager, W.: "An Introduction to Plasticity," AddisonWesley Publishing Company,
Reading, Mass., 1959.
and P. G. Hodge, Jr. : "Theory of Perfectly Plastic Solids," John Wiley & Sons,
Inc., New York, 1951.
1 R. Hill, Quart. J. Mech. Appl. Math, vol. 1, pp. 4052, 1949.
2 J. M. Alexander, Proc. Inst. Mech. Engrs., (London), vol. 169, pp. 10211030, 1955.
* W. Prager, Trans. Roy. Inst. TechnoL, Stockholm, no. 65, 1953.
^E. C. Thomsen, /. Appl. Mech., vol. 24, pp. 8184, 1957.
" B. B. Hundy, Meiallurgia, vol. 49, no. 293, pp. 109118, 1954.
Part Two
METALLURGICAL FUNDAMENTALS
Chapter 4
PLASTIC DEFORMATION
OF SINGLE CRYSTALS
41 . Introduction
The previous three chapters have been concerned with the phenomeno
logical description of the elastic and plastic behavior of metals. It has
been shown that formal mathematical theories have been developed for
describing the mechanical behavior of metals based upon the simplifying
assumptions that metals are homogeneous and isotropic. That this is
not true should be obvious to anyone who has examined the structure
of metals under a microscope. However, for finegrained metals sub
jected to static loads within the elastic range the theories are perfectly
adequate for design. Within the plastic range the theories describe the
observed behavior, although not with the precision which is frequently
desired. For conditions of dynamic and impact loading we are forced,
in general, to rely heavily on experimentally determined data. As the
assumption that we are dealing with an isotropic homogeneous medium
becomes less tenable, our ability to predict the behavior of metals under
stress by means of the theories of elasticity and plasticity decreases.
Following the discovery of the diffraction of X rays by metallic crystals
by Von Laue in 1912 and the realization that metals were fundamentally
composed of atoms arranged in specific geometric lattices there have been
a great many investigations of the relationships between atomic structure
and the plastic behavior of metals. Much of the fundamental work on
the plastic deformation of metals has been performed with singlecrystal
specimens, so as to eliminate the complicating effects of grain boundaries
and the restraints imposed by neighboring grains and secondphase parti
cles. Techniques for preparing single crystals have been described in a
number of sources.^"*
' R. W. K. Honeycombe, Met. Reviews, vol. 4, no. 13, pp. 147, 1959.
2 A. N. Holden, Trans. ASM, vol. 42, pp. 319346, 1950.
3 W. D. Lawson and S. Nielsen, "Preparation of Single Crystals," Academic Press,
Inc., New York, 1958.
81
82 Metallurgical Fundamentals
[Ch
ap.
The basic mechanisms of plastic deformation in single crystals will be
discussed in this chapter. This subject will be extended in the next chap
ter to a consideration of plastic deformation in polycrystalline specimens.
Primary consideration will be given to tensile deformation. The funda
mental deformation behavior in creep and fatigue will be covered in
chapters specifically devoted to these subjects. The dislocation theory,
which plays such an important part in presentday concepts of plastic
deformation, will be introduced in this chapter to the extent needed to
provide a qualitative understanding of modern concepts of plastic defor
mation. A more detailed consideration of dislocation theory will be
found in Chap. 6. This will be followed by a chapter on the funda
mental aspects of fracture and a chapter on internal friction and anelastic
effects.
42. Concepts oF Crystal Geometry
Xray diffraction analysis shows that the atoms in a metal crystal are
arranged in a regular, repeated threedimensional pattern. The atom
arrangement of metals is most simply portrayed by a crystal lattice in
which the atoms are visualized as hard balls located at particular locations
in a geometrical arrangement.
The most elementary crystal structure is the simple cubic lattice (Fig.
41). This is the type of structure cell found for ionic crystals, such as
NaCl and LiF, but not for any oi'
the metals. Three mutually perpen
dicular axes are arbitrarily placed
through one of the corners of the cell.
Crystallographic planes and direc
tions will be specified with respect to
these axes in terms of Miller indices.
A crystallographic plane is specified
in terms of the length of its inter
cepts on the three axes, measured
from the origin of the coordinate axes.
To simplify the crystallographic
formulas, the reciprocals of these in
tercepts are used. They are reduced
to a lowest common denominator to give the Miller indices (hkl) of the
plane. For example, the plane ABCD in Fig. 41 is parallel to the x and
z axes and intersects the y axis at one interatomic distance ao. There
fore, the indices of the plane are 1/ oo , 1/1, 1/ oo , or (hkl) = (010). Plane
EBCF would be designated as the (TOO) plane, since the origin of the
coordinate system can be moved to G because every point in a space
Fig. 41 . Simple cubic structure.
42]
Plastic Deformation of Single Crystals 83
[\00) HADG
[WO] HBCG
[\\\)GEC
[\\2)GJC
lattice has the same arrangement of points as every other point. The
bar over one of the integers indicates that the plane intersects one of the
axes in a negative direction. There are six crystallographically equiva
lent planes of the type (100), any one of which can have the indices (100),
(010), (001), (TOO), (OlO), (OOT) depending upon the choice of axes. The
notation J 100} is used when they
are to be considered as a group,
or family of planes.
Crystallographic directions are
indicated by integers in brackets:
[uvw]. Reciprocals are not used in
determining directions. As an ex
ample, the direction of the line F D
is obtained by moving out from the
origin a distance ao along the x axis
and moving an equal distance in the
positive y direction. The indices
of this direction are then [110].
A family of crystallographically
equivalent directions would be des
ignated {uvw.) For the cubic lat
tice only, a direction is always
perpendicular to the plane having
the same indices.
Many of the common metals have
either a bodycentered cubic (bcc) or
facecentered cubic (fee) crystal struc
ture. Figure 42a shows a body
centered cubic structure cell with
an atom at each corner and another
atom at the body center of the cube.
Each corner atom is surrounded
by eight adjacent atoms, as is the
atom located at the center of the cell.
Fig. 42. (a) Bodycentered cubic struc
ture; (6) facecentered cubic structure.
Therefore, there are two atoms per structure cell for the bodycentered
cubic structure {% \ I). Typical metals which have this crystal struc
ture are alpha iron, columbium, tantalum, chromium, molybdenum, and
tungsten. Figure 426 shows the structure cell for a facecentered cubic
crystal structure. In addition to an atom at each corner, there is an
atom at the center of each of the cube faces. Since these latter atoms
belong to two unit cells, there are four atoms per structure cell in the
facecentered cubic structure (% f ^^). Aluminum, copper, gold, lead,
silver, and nickel are common facecentered cubic metals.
84
Metallurgical Fundamentals
[Ch
ap.
The third common metallic crystal structure is the hexagonal close
packed (hep) structure (Fig. 43). In order to specify planes and direc
tions in the hep structure, it is convenient to use the MillerBravais
system with four indices of the type (hkil). These indices are based on
four axes ; the three axes ai, a^, az are
120° apart in the basal plane, and
the vertical c axis is normal to the
basal plane. These axes and typical
planes in the hep crystal structure
are given in Fig. 43. The third
index is related to the first two by
the relation i = —{h\k).
Basal plane (0001)  ABCDEF
Prism plane (1010)  FEJH
Pyramidal planes
Type I, Order 1 (lOTl)  GHJ
Type I, Order 2 (101_2)  KJH
Type n, Order 1 (1 1 21 )  GHL
Type n, Order 2 (1122)  KHL
Digonal axis [ll20]  FGC
Fig. 43.
ture.
Hexagonal closepacked struc Fig. 44. Stacking of closepacked spheres.
The facecentered cubic and hexagonal closepacked structures can both
be built up from a stacking of closepacked planes of spheres. Figure 44
shows that there are two ways in which the spheres can be stacked. The
first layer of spheres is arranged so that each sphere is surrounded by
and just touching six other spheres. This corresponds to the solid circles
in Fig. 44. A second layer of closepacked spheres can be placed over
the bottom layer so that the centers of the atoms in the second plane
cover onehalf the number of valleys in the bottom layer (dashed circles
in Fig. 44). There are two ways of adding spheres to give a third close
packed plane. Although the spheres in the third layer must fit into the
valleys in the second plane, they may lie either over the valleys not
covered in the first plane (the dots in Fig. 44) or directly above the
atoms in the first plane (the crosses in Fig. 44). The first possibility
results in a stacking sequence ABC ABC ■ • • , which is found for the
Sec. 43] Plastic Deformation of Single Crystals 85
{111} planes of an fee strueture. The other possibility results in the stack
ing sequence ABAB • ■ , which is found for the (0001) basal plane of
the hep strueture. For the ideal hep packing, the ratio of c/a is \/%,
or 1.633. Table 41 shows that actual hep metals deviate from the ideal
c/a ratio.
Table 41
Axial Ratios of Some Hexagonal Metals
Metal c/a
Be 1.568
Ti 1.587
Mg 1.623
Ideal hep 1 . 633
Zn 1 . 856
Cd 1 . 886
The fee and hep structures are both closepacked structures. Seventy
four per cent of the volume of the unit cell is occupied by atoms, on a
hard sphere model, in the fee and hep structures. This is contrasted
with 68 per cent packing for a bcc unit cell and 52 per cent of the volume
occupied by atoms in the simple cubic unit cell.
Plastic deformation is generally confined to the lowindex planes, which
have a higher density of atoms per unit area than the highindex planes.
Table 42 lists the atomic density per unit area for the common lowindex
planes. Note that the planes of greatest atomic density also are the
most widely spaced planes for the crystal structure.
Table 42
Atomic Density of Lowindex Planes
Atomic density,
Distance
Crystal structure
Plane
atoms per
between
unit area
planes
Facecentered cubic
Octahedral lllj
4/v/3 ao^
ao/V3
ao/2
Cube {1001
2/ao^
Dodecahedral {1101
2/^/2 ao^
ao/2 V2
Bodycentered cubic
Dodecahedral {llOj
2/ V2 oo^
ao/V2
Cube {1001
l/flo^
Oo/2
Octahedral {1111
l/V3ao2
ao/2 y/3
Hexagonal closepacked
Basal {00011
1/3 V3ao2
c
43. Lattice Defects
Real crystals deviate from the perfect periodicity that was assumed in
the previous section in a number of important ways. While the concept
86 Metallurgical Fundamentals [Chap. 4
of the perfect lattice is adequate for explaining the structureinsensitive
properties of metals, for a better understanding of the structuresensitive
properties it has been necessary to consider a number of types of lattice
defects. The description of the structuresensitive properties then
reduces itself largely to describing the behavior of these defects.
Structureinsensitive Structuresensitive
Elastic constants Electrical conductivity
Melting point Semiconductor properties
Density Yield stress
Specific heat Fracture strength
CoeflScient of thermal expansion Creep strength
As is suggested by the above brief tabulation, practically all the
mechanical properties are structuresensitive properties. Only since the
realization of this fact, in relatively recent times, have really important
advances been made in understanding the mechanical behavior of
materials.
The term defect, or imperfection, is generally used to describe any devi
ation from an orderly array of lattice points. When the deviation from
the periodic arrangement of the lattice is localized to the vicinity of only
a few atoms it is called a point defect, or point imperfection. However,
if the defect extends through microscopic regions of the crystal, it is
called a lattice imperfection. Lattice imperfections may be divided into
line defects and surface, or plane, defects. Line defects obtain their name
because they propagate as lines or as a twodimensional net in the crystal.
The edge and screw dislocations that are discussed in this section are the
common line defects encountered in metals. Surface defects arise from
the clustering of line defects into a plane. Lowangle boundaries and
grain boundaries are surface defects (see Chap. 5). The stacking fault
between two closepacked regions of the crystal that have alternate stack
ing sequences (Sec. 410) is also a surface defect.
Point Defects
Figure 45 illustrates three types of point defects. A vacancy, or vacant
lattice site,^ exists when an atom is missing from a normal lattice position
(Fig. 45a). In pure metals, small numbers of vacancies are created by
thermal excitation, and these are thermodynamically stable at temper
atures greater than absolute zero. At equilibrium, the fraction of lattices
that are vacant at a given temperature is given approximately by the
equation
n lis /A i\
^ = exp^ (41)
1 '
'Vacancies and Other Point Defects," Institute of Metals, London, 1958.
bee. 4ii Plastic Deformation of Single Crystals 87
where n is the number of vacant sites in N sites and E^ is the energy
required to move an atom from the interior of a crystal to its surface.
Table 43 illustrates how the fraction of vacant lattice sites in a metal
increases rapidly with temperature. By rapid quenching from close to
the melting point, it is possible to trap in a greater than equilibrium
oooo oooo ooooo
oooo oooo ooooo
oo o oooo o^ooo
o
oooo oooo ooooo
ia) [b] [c)
Fig. 45. Point defects, (a) Vacancy; (6) interstitial; (c) impurity atom.
number of vacancies at room temperature. Higher than equilibrium
concentrations of vacancies can also be produced by extensive plastic
deformation (cold work) or as the result of bombardment with high
energy nuclear particles. When the density of vacancies becomes rela
tively large, it is possible for them to cluster together to form voids.
Table 43
Equilibrium Vacancies in a Metal
Approximate fraction
Temperature, °C
of vacant lattice sites
500
1 X 1010
1000
1 X 106
1500
5 X 10"
2000
3 X 103
E, ~ 1 ev
An atom that is trapped inside the crystal at a point intermediate
between normal lattice positions is called an interstitial atom, or inter
stitialcy (Fig. 456). The interstitial defect occurs in pure metals as a
result of bombardment with highenergy nuclear particles (radiation
damage), but it does not occur frequently as a result of thermal activation.
The presence of an impurity atom at a lattice position (Fig. 45c) or at
an interstitial position results in a local disturbance of the periodicity of
the lattice, the same as for vacancies and interstitials.
Line Defects — Dislocations
The most important twodimensional, or line, defect is the dislocation.
The dislocation is the defect responsible for the phenomenon of slip, by
which most metals deform plastically. Therefore, one way of thinking
about a dislocation is to consider that it is the region of localized lattice
88 Metallurgical Fundamentals [Chap. 4
disturbance separating the slipped and unslipped regions of a crystal.
In Fig. 46, AB represents a dislocation lying in the slip plane, which is
the plane of the paper. It is assumed that slip is advancing to the right.
All the atoms above area C have been displaced one atomic distance in
the slip direction; the atoms above D have not yet slipped. AB is then
the boundary between the slipped and unslipped regions. It is shown
shaded to indicate that for a few atomic dis
tances on each side of the dislocation line
there is a region of atomic disorder in which
the slip distance is between zero and one
atomic spacing. As the dislocation moves,
slip occurs in the area over which it moves.
In the absence of obstacles, a dislocation can
move easily on the application of only a small
force; this helps explain why real crystals
.,,,., . . deform much more readily than would be
Fig. 46. A dislocation in a ^ i r ^ i ^i r , ■, ,.•
slip plane expected tor a crystal with a perfect lattice.
Not only are dislocations important for ex
plaining the slip of crystals, but they are also intimately connected with
nearly all other mechanical phenomena such as strain hardening, the
yield point, creep, fatigue, and brittle fracture.
The two basic types of dislocations are the edge dislocation and the
screw dislocation. The simplest type of dislocation, which was originally
suggested by Orowan, Polanyi, and Taylor, is called the edge disloca
tion, or TaylorOrowan dislocation. Figure 47 shows the slip that
produces an edge dislocation for an element of crystal having a simple
cubic lattice. Slip has occurred in the direction of the slip vector over
the area ABCD. The boundary between the righthand slipped part of
the crystal and the lefthand part which has not yet slipped is the line
AD, the edge dislocation. Note that the parts of the crystal above the
slip plane have been displaced, in the direction of slip, with respect to
the part of the crystal below the slip plane by an amount indicated by
the shaded area in Fig. 47. All points in the crystal which were origi
nally coincident across the slip plane have been displaced relative to each
other by this same amount. The amount of displacement is equal to the
Burgers vector b of the dislocation. For a pure edge dislocation such as is
shown here, the magnitude of the Burgers vector is equal to the atomic
spacing. A defining characteristic of an edge dislocation is that its
Burgers vector is always perpendicular to the dislocation line.
Although the exact arrangement of atoms along AZ) is not known, it is
generally agreed that Fig. 48 closely represents the atomic arrangement
in a plane normal to the edge dislocation AD. The plane of the paper in
this figure corresponds to a (100) plane in a simple cubic lattice and is
Sec. 43]
Plastic Deformation of Single Crystals
89
equivalent to any plane parallel to the front face of Fig. 47. Note that
the lattice is distorted in the region of the dislocation. There is one more
vertical row of atoms above the slip plane than below it. The atomic
arrangement results in a compressive stress above the slip plane and a
tensite stress below the slip plane. An edge dislocation with the extra
plane of atoms above the slip plane, as in Fig. 48, is called by convention
a positive edge dislocation and is frequently indicated by the symbol X.
Slip vector
vector
Fig. 47. Edge dislocation produced by
slip in a simple cubic lattice. Disloca
tion lies along AD, perpendicular to slip
direction. Slip has occurred over area
ABCD. {W. T. Read, Jr., "Disloca
tions in Crystals," p. 2, McGrawHill
Book Company, Inc., New York, 1953.)
■' ,, r .
r T ? ? 9
J^Xii
rzijirrn .
1
Hz ^rm
A 4 4 A i i i A
Fig. 48. Atomic arrangement in a plane
normal to an edge dislocation. {W. T.
Read, Jr., "Dislocations in Crystals," p. 3,
]\IcGrawHill Book Company, Inc., New
York, 1953.)
If the extra plane of atoms lies below the slip plane, the dislocation is a
negative edge dislocation, T .
A pure edge dislocation can glide or slip in a direction perpendicular
to its length. However, it may move vertically by a process known as
climb, if diffusion of atoms or vacancies can take place at an appreciable
rate. Consider Fig. 48. For the edge dislocation to move upward
(positive direction of climb), it is necessary to remove the extra atom
directly over the symbol ± or to add a vacancy to this spot. One such
atom would have to be removed for every atomic spacing that the dis
location climbs. Conversely, if the dislocation moved down, atoms
would have to be added. Atoms could be removed from the extra
plane of atoms by the extra atom interacting with a lattice vacancy.
Atoms are added to the extra plane for negative climb by the diffusion
of an atom from the surrounding crystal, creating a vacancy. Since
movement by climb is diffusioncontrolled, motion is much slower than
in glide and less likely except at high temperatures.
90
Metallurgical Fundamentals
[Ch
ap.
The second basic type of dislocation is the screw, or Burgers, dislocation,
J'igure 49 shows a simple example of a screw dislocation. The upper
part of the crystal to the right oi AD has moved relative to the lower
part in the direction of the slip vector. No slip has taken place to the
left of AD, and therefore AD is a, dislocation line. Thus, the dislocation
line is parallel to its Burgers vector, or slip vector, and by definition this
must be a screw dislocation. Consider the trace of a circuit around the
dislocation line, on the front face of the crystal. Starting at X and com
pleting the circuit, we arrive at X',
one atomic plane behind that con
taining X. In making this circuit
we have traced the path of a right
handed screw. Every time a cir
cuit is made around the dislocation
line, the end point is displaced one
plane parallel to the slip plane in
the lattice. Therefore, the atomic
planes are arranged around the
dislocation in a spiral staircase or
screw.
The arrangement of atoms (in
two dimensions) around a screw
dislocation in a simple cubic lattice
is shown in Fig. 410. In this figure
we are looking down on the slip
plane in Fig. 49. The open circles
represent atoms just above the slip
plane, and the solid circles are atoms just below the slip plane. A screw
dislocation does not have a preferred slip plane, as an edge dislocation
has, and therefore the motion of a screw dislocation is less restricted than
the motion of an edge dislocation. However, movement by climb is
not possible with a screw dislocation.
For the present, the discussion of dislocations will be limited to the
geometrical concepts presented in this section. After a more complete
discussion of the plastic deformation of single crystals and polycrystalline
specimens, we shall return to a detailed discussion of dislocation theory in
Chap. 6. Among the topics covered will be the effect of crystal structure
on dislocation geometry, the experimental evidence for dislocations, and
the interaction between dislocations.
/'Slip
^ vector
Fig. 49. Slip that produces a screw dis
location in a simple cubic lattice. Dis
location lies along AD, parallel to slip
direction. Slip has occurred over the
area ABCD. (W. T. Read, Jr., "Disloca
tions in Crystals," p. 15, McGrawHill
Book Company, Inc., New York, 1953.)
44. Deformation by Slip
The usual method of plastic deformation in metals is by the sliding of
blocks of the crystal over one another along definite crystallographic
Sec. 44]
Plastic Deformation of Single Crystals
91
planes, called slip planes. As a very crude approximation, the slip, or
glide, of a crystal can he considered analogous to the distortion produced
in a deck of cards when it is pushed from one end. Figure 411 illus
trates this classical picture of slip. In Fig. 41 la, a shear stress is applied
® — ® — ® — ® — ® — ®=^
<ii 6) ii ii ^i «>=^
<i^ «) «5 6) «J 6N^
«? 65 ^) 6) ^)
5 6 J «5 i) 65 6^
(5 65 6 5 ^5 6 5 6^
65 65
$ 65 65 65 6) 6^
65 65 65 65 65 6 5=:^
(5 6) 65 65 6^
(5 65 ® 6^
«5 65 6^
(5 65 6 5 6 5 ^
65 65 6) 6) ®
6 5 ® 6 5 6 5 65
6 5 65 65 6 5 65 6>=.
65 6 5 6 5 65 65 6^
® ® ® ® ®
^5 65 6) 65 6^
Slip
vector
Fig. 410. Atomic arrangement around the screw dislocation shown in Fig. 49. The
plane of the figure is parallel to the slip plane. ABCD is the slipped area, and AD is
the screw dislocation. Open circles represent atoms in the atomic plane just above
the slip plane, and the solid circles are atoms in the plane just below the slip plane.
(W. T. Read, Jr., "Dislocations in Crystals," p. 17, McGrawHill, Book Company, Inc.,
New York, 1953.)
to a metal cube with a top polished surface. Slip occurs when the shear
stress exceeds a critical value. The atoms move an integral number of
atomic distances along the slip plane, and a step is produced in the
polished surface (Fig. 4116). When we view the polished surface from
above with a microscope, the step shows up as a line, which w^e call a
slip line. If the surface is then repolished after slip has occurred, so that
the step is removed, the slip line will disappear (Fig. 41 Ic). The single
92
Metallursical Fundamentals
[Chap. 4
crystal is still a single crystal after slip has taken place provided that
the deformation was uniform. Note that slip lines are due to changes in
surface elevation and that the surface must be suitably prepared for
Polished surface
• • • 1 • • •
^
Slip
ine
r
1'
Slip
plane
U) {^) (c)
Fig. 41 1 . Schematic drawing of classical idea of slip.
Fig. 412. Straight slip lines in copper, 500 X. {Courtesy W. L. Phillips.)
microscopic observation prior to deformation if the slip lines are to be
observed. Figure 412 shows straight slip lines in copper.
The fine structure of slip lines has been studied at high magnification
by means of the electron microscope. What appears as a line, or at best
a narrow band, at 1,500 diameters' magnification in the optical microscope
can be resolved by the electron microscope as discrete slip lamellae at
20,000 diameters, shown schematically in Fig. 413. The situation where
Sec. 44]
Plastic Deformation of Single Crystals
93
there are many slip lamellae comprising the slip band is found for alumi
num and copper, but in alpha brass there is only one slip line, even when
viewed at high magnification.
Slip occurs most readily in specific directions on certain crystallographic
planes. Generally the slip plane is the plane of greatest atomic density
(Table 42), and the slip direction is the closestpacked direction within
the slip plane. Since the planes of greatest atomic density are also the
most widely spaced planes in the crystal structure, the resistance to slip
is generally less for these planes than for any other set of planes. The
slip plane together with the slip direction establishes the slip system.
Slip distance
^Interslip
/ region
Lamella
spacing
id)
Fig. 41 3. Schematic drawing of the fine structure of a sKp band,
tion; (6) large deformation.
(a) Small deforma
In the hexagonal closepacked metals, the only plane with high atomic
density is the basal plane (0001). The digonal axes (1120) are the close
packed directions. For zinc, cadmium, magnesium, and cobalt slip
occurs on the (0001) plane in the (1120) directions.^ Since there is only
one basal plane per unit cell and three (1120) directions, the hep structure
possesses three slip systems. The limited number of slip systems is the
reason for the extreme orientation dependence of ductility in hep crystals.
In the facecentered cubic structure, the {lllj octahedral planes and
the (110) directions are the closepacked systems. There are eight  111 }
planes in the fee unit cell. However, the planes at opposite corners of
the cube are parallel to each other, so that there are only four sets of
octahedral planes. Each { 111 } plane contains three (110) directions (the
reverse directions being neglected). Therefore, the fee lattice has 12
possible slip systems.
The bcc structure is not a closepacked structure like the fee or hep
structures. Accordingly, there is no one plane of predominant atomic
density, as (111) in the fee structure and (0001) in the hep structure.
The {110} planes have the highest atomic density in the bcc structure,
' Zirconium and titanium, which have low c/a ratios, slip primarily on the prism and
pyramidal planes in the (1120) direction.
94 Metallurgical Fundamentals [Chap. 4
but they are not greatly superior in this respect to several other planes.
However, in the bee structure the (111) direction is just as closepacked
as the (110) and (1120) directions in the fee and hep structures. There
fore, the bcc metals obey the general rule that the slip direction is the
closepacked direction, but they differ from most other metals by not
having a definite single slip plane. Slip in bcc metals is found to occur
on the {110}, {112}, and {123} planes, while the slip direction is always
the [111] direction. There are 48 possible slip systems, but since the
€P
Fig. 414. Wavy alip lines in alpha iron, 150 X. {Cuuiicsy J. J. Cox.)
planes are not so closepacked as in the fee structure, higher shearing
stresses are usually required to cause slip.
Slip in bcc alpha iron has been particularly well studied.^ It has been
concluded that the slip plane in alpha iron may occupy any position in
the [111] zone, its position being determined by the orientation of the
stress axis with respect to the crystal axis and the variation in the shear
ing strengths of the planes in the slip zone. These studies have shown
that observed deviations from the lowindex planes {110}, {112}, and
{ 123} are real effects, which supports the belief that slip in alpha iron is
noncrystallographic. Further evidence for noncrystallographic slip is the
fact that slip lines in alpha iron are wavy^ (Fig. 414).
Certain metals show additional slip systems with increased temper
1 F. L. Vogel and R. M. Brick, Trans. AIME, vol. 197, p. 700, 1958; R. P. Steijn
and R. M. Brick, Trans. ASM, vol. 4G, pp. 140G1448, 1954; J. J. Cox, G. T. Home,
and R. F. MchI, Trans. ASM, vol. 49, 118131, 1957.
2 J. R. Low and R. W. Guard, Acta Mel., vol. 7, pp. 171179, 1959, have shown that
curved slip linos are produced in iron by screw components of the dislocation loop
but that the slip lines are straight when viewed normal to the edge component of the
dislocation.
Sec. 45]
Plastic Deformation of Single Crystals
95
ature. Aluminum deforms on 'he {100} plane at elevated temperature,
while in magnesium the {lOllj pyramidal plane plays an important role
in deformation by slip above 225°C. In all cases the slip direction
remains the same when the slip plane changes with temperature.
45. Slip in a Perfect Lattice
If slip is assumed to occur by the translation of one plane of atoms
over another, it is possible to make a reasonable estimate^ of the shear
o o o o
o o o
[a)
/
^Sinusoidal relationsh
^/■~\ \^^ Realistic relations
p
hip
Displacement x
Fig. 415. (a) Shear displacement of one plane of atoms over another atomic plane;
(6) variation of shearing stress with displacement in sHp direction.
stress required for such a movement in a perfect lattice. Consider two
planes of atoms subjected to a homogeneous shear stress (Fig. 415).
The shear stress is assumed to act in the slip plane along the slip direc
tion. The distance between atoms in the slip direction is b, and the
spacing between adjacent lattice planes is a. The shear stress causes a
displacement x in the slip direction between the pair of adjacent lattice
planes. The shearing stress is initially zero when the two planes are in
coincidence, and it is also zero when the two planes have moved one
identity distance h, so that point 1 in the top plane is over point 2 on the
bottom plane. The shearing stress is also zero when the atoms of the
top plane are midway between those of the bottom plane, since this is a
' J. Frenkel, Z. Physik, vol. 37, p. 572, 1926.
96 Metallurgical Fundamentals [Chap. 4
symmetry position. Between these positions each atom is attracted
toward the nearest atom of the other row, so that the shearing stress is
a periodic function of the displacement.
As a first approximation, the relationship between shear stress and dis
placement can be expressed by a sine function
T = Tm Sni —r ' (42)
where r^ is the amplitude of the sine wave and h is the period. At small
values of displacement, Hooke's law should apply.
Gx
r = Gy =~ (43)
For small values of x/h, Eq. (42) can be written
27ra;
(44)
Combining Eqs. (43) and (44) provides an expression for the maximum
shear stress at which slip should occur.
As a rough approximation, h can be taken equal to a, with the result that
the theoretical shear strength of a perfect crystal is approximately equal
to the shear modulus divided by 2x.
Tm^§^ (46)
The shear modulus for metals is in the range 10® to 10'^ psi (10^^ to
10^2 dynes/cm^). Therefore, Eq. (46) predicts that the theoretical shear
stress will be in the range 10^ to 10® psi, while actual values of the shear
stress required to produce plastic deformation are in the range 10^ to
10* psi. Even if more refined calculations are used to correct the sine
wave assumption, the value of r^ cannot be reduced by more than a
factor of 5 from the value predicted by Eq. (46). Thus, it seems reason
able to expect that the theoretical shear strength of most metals lies
between G/10 and G/50. This is still at least 100 times greater than the
observed shear strengths of metal crystals. It can only be concluded
that a mechanism other than the bodily shearing of planes of atoms is
responsible for slip. In the next section, it is shown that dislocations
provide such a mechanism.
Sec. 46] Plastic Deformation of Single Crystals 97
46. Slip by Dislocation Movement
The concept of the dislocation was first introduced to explain the dis
crepancy between the observed and theoretical shear strengths of metals.
For the concept to be useful in this field, it is necessary to demonstrate
(1) that the passage of a dislocation through a crystal lattice requires
far less than the theoretical shear stress and (2) that the movement of
the dislocation through the lattice produces a step, or slip band, at the
free surface.
1 23456789
id)
Fig. 416. Schematic diagram illustrating the fact that a dislocation moves easily
through a crystal lattice, (a) Energy field in perfect crystal lattice; (6) lattice
containing an edge dislocation. {F. Seilz, "The Physics of Metals," p. 91, McGraw
Hill Book Company, Inc., New York, 1943.)
To illustrate that the stress required to move a dislocation through a
crystal is very low compared with the theoretical shear stress, we shall
use Fig. 416. Figure 416a represents the atoms in two adjacent planes
in a perfect crystal lattice which does not contain a dislocation. The
top curve of the figure represents schematically the energy of an atom in
the lower plane of atoms as a function of its position relative to the upper
plane. For the normal arrangement of a perfect crystal, all the atoms
in the lower plane are at minimum positions in the energy curve. There
fore, if the top row of atoms is displaced toward the right relative to the
bottom row, each atom encounters the same force opposing the displace
ment. This is the situation described in Sec. 45. Now consider the
situation when the crystal contains a dislocation (Fig. 4166). This illus
trates a positive edge dislocation, with the extra plane of atoms between
4 and 5. The atoms at large distances from the center of the dislocation
are at positions corresponding to the minimum of the energy curve; the
atoms at the center are not. Now consider pairs of atoms, for example.
98 Metallurgical Fundamentals
rch
ap.
4 and 5, 3 and 6, etc., located symmetrically on opposite sides of the
center of the dislocation. They encounter forces which are equal and
opposite. As a result, if the atoms near the center of the dislocation are
displaced by equal distances, onehalf will encounter forces opposing the
motion and onehalf will encounter forces which assist the motion.
Therefore, to a first approximation, the net work required to produce
Fig. 41 7. Movement of edge dislocation in a simple cubic lattice.
Roy. Soc. (London), vol. 145A, p. 369, 1934.)
(G. I. Taylor, Proc.
the displacement is zero, and the stress required to move the dislocation
one atomic distance is very small.
The lattice offers essentially no resistance to the motion of a dislocation
only when the dislocation lies at a position of symmetry with respect to
the atoms in the slip plane. In general, a small force, the PeierlsNabarro
force, is needed to drive a dislocation through the lattice. While it is
well established that the value of the PeierlsNabarro force is much
smaller than the theoretical shear stress for a perfect lattice, the accu
rate calculation of this force is difficult because it depends strongly on
the relatively uncertain atomic arrangement at the center of a dislocation.
Figure 417, based on the original work by Taylor,^ illustrates that the
movement of a dislocation results in a surface step, or slip band. The
top series of sketches shows a positive edge dislocation moving to the right
in a simple cubic lattice. The slip plane is shown dashed. When the dis
location reaches the right side of the crystal, assumed to be a free surface,
' G. I. Taylor, Proc. Roy. Soc. (London), vol. 145A, p. 362, 1934.
)ec,
47]
Plastic Deformation of Single Crystals 99
it produces a shift with respect to the planes on each side of the slip plane
of one Burgers vector, or one atomic distance for the simple cubic lattice.
The bottom series of sketches shows that the same surface step is pro
duced by the movement of a negative edge dislocation to the left.
Slip
direction
Slip plane
47. Critical Resolved Shear Stress for Slip
The extent of slip in a single crystal depends on the magnitude of the
shearing stress produced by external loads, the geometry of the crystal
structure, and the orientation of the
active slip planes with respect to the
shearing stresses. Slip begins when
the shearing stress on the slip plane in
the slip direction reaches a threshold
value called the critical resolved shear
stress. This value ^ is really the single
crystal equivalent of the yield stress of
an ordinary stressstrain curve. The
value of the critical resolved shear
stress depends chiefly on composition
and temperature.
The fact that different tensile loads
are required to produce slip in single
crystals of different orientation can be
rationalized by a critical resolved shear
stress; this w^as first recognized by
Schmid. To calculate the critical re
solved shear stress from a single crystal
tested in tension, it is necessary to know, from Xray diffraction, the
orientation with respect to the tensile axis of the plane on which slip first
appears and the slip direction. Consider a cylindrical single crystal with
crosssectional area A (Fig. 418). The angle between the normal to the
slip plane and the tensile axis is 0, and the angle which the slip direc
tion makes with the tensile axis is X. The area of the slip plane inclined
at the angle 4> will be A /cos 0, and the component of the axial load
acting in the slip plane in the slip direction is P cos X. Therefore, the
critical resolved shear stress is given by
Fig. 41 8. Diagram for calculating
critical resolved shear stress.
P COS X P
'■^ ^ ~T7? 1\ = "T COS (^ COS X
A/(cos <t>) A
(47)
1 In practice it is very difficult to determine the stress at which the first slip bands
are produced. In most cases, the critical shear stress is obtained by the intersection
of the extrapolated elastic and plastic regions of the stressstrain curve.
2 E. Schmid, Z. Elektrochem., vol. 37, p. 447, 1931.
100 Metallurgical Fundamentals
[Ch
ap.
The law of the critical resolved shear stress, also known as Schmid's
law, is best demonstrated with hep metals, where the limited number of
slip systems allows large differences in orientation between the slip plane
and the tensile axis (see Prob. 48). In fee metals the high symmetry
results in so many equivalent slip systems that it is possible to get a
Table 44
Roomtemperature Slip Systems and Critical Resolved Shear Stress
FOR Metal Single Crystals
Metal
Zn.
Mg
Cd.
Ti.
Ag.
Cu.
Ni.
Fe.
Mo
Crystal
structure
hep
hep
hep
hep
fee
fee
fee
bee
bee
Purity,
%
99.999
99.996
99.996
99.99
99.9
99.99
99.97
99.93
99.999
99.98
99.8
99.96
Slip
plane
(0001)
(0001)
(0001)
(1010)
(1010)
(111)
(111)
(111)
(111)
(111)
(111)
(110)
(112)
(123)
(110)
Slip
direction
[1120]
[1120]
[1120]
[1120]
[1120]
[110]
[110]
[110]
[110]
[110]
[110]
[111]
[111]
Critical
shear stress,
g/mm^
18
77
58
1,400
9,190
48
73
131
65
94
580
2,800
5,000
Ref.
« D. C. Jillson, Trans. AIME, vol. 188, p. 1129, 1950.
^ E. C. Burke and W. R. Hibbard, Jr., Trans. AIME, vol. 194, p. 295, 1952.
'^ E. Schmid, "International Conference on Physics," vol. 2, Physical Society,
London, 1935.
<* A. T. Churchman, Proc. Roy. Soc. (London), vol. 226A, p. 216, 1954.
« F. D. Rosi, Trans. AIME, vol. 200, p. 1009, 1954.
/J. J. Cox, R. F. Mehl, and G. T. Home, Trans. ASM, vol. 49, p. 118, 1957.
"R. Maddin and N. K. Chen, Trans. AIME, vol. 191, p. 937, 1951.
variation in the yield stress of only about a factor of 2 because of differ
ences in the orientation of the slip plane with the tensile axis. The
demonstration of the resolvedshearstress law is even less favorable in
bcc metals owing to the large number of available slip systems. How
ever, available data indicate that Schmid's law is obeyed for cubic metals
as well as hep metals.
Table 44 gives values of critical resolved shear stress for a number of
metals. The importance of small amounts of impurities in increasing
the critical resolved shear stress is shown by the data for silver and copper.
Sec. 47]
Plastic Deformation of Single Crystals
101
Alloyingelement additions have even a greater effect, as shown by the
data for goldsilver alloys in Fig. 419. Note that a large increase in the
resistance to slip is produced by alloying gold and silver even though
these atoms are very much alike in size and electronegativity, and hence
Ag
20
00
Au
40 60
Atom % Au
Fig. 419. Variation of critical resolved shear stress with composition in silvergold
alloy single crystals. ((?. Sachs and J. Weerts, Z. Physik, vol. 62, p. 473, 1930.)
28
24
20
16
2? 12
I
8
J
\
o \
l\
t \
• N —
N. °
250 200
150
100 50
Temoerature, °C
50
100
150
.CO
Fig. 420. Variation of critical resolved shear stress with temperature for iron single
crystals. (J. J. Cox, R. F. Mehl, and G. T. Home, Trans. ASM, vol. 49, p. 123, 1957.)
they form a solid solution over the complete range of composition. In
solid solutions, where the solute atoms differ considerably in size from the
solvent atoms, an even greater increase in critical resolved shear stress
would be observed.
102 Metallurgical Fundamentals [Chap. 4
The magnitude of the critical resolved shear stress of a crystal is deter
mined by the interaction of its population of dislocations with each other
and with defects such as vacancies, interstitials, and impurity atoms.
This stress is, of course, greater than the stress required to move a single
dislocation, but it is appreciably lower than the stress required to pro
duce slip in a perfect lattice. On the basis of this reasoning, the critical
resolved shear stress should decrease as the density of defects decreases,
provided that the total number of imperfections is not zero. When the
last dislocation is eliminated, the critical resolved shear stress should rise
abruptly to the high value predicted for the shear strength of a perfect
crystal. Experimental evidence for the effect of decreasing defect den
sity is shown by the fact that the critical resolved shear stress of soft
metals can be reduced to less than onethird by increasing the purity. At
the other extreme, microndiameter singlecrystal filaments, or whiskers,
can be grown essentially dislocationfree. Tensile tests ^ on these fila
ments have given strengths which are approximately equal to the calcu
lated strength of a perfect crystal.
48, Testing of Single Crystals
Most studies of the mechanical properties of single crystals are made
by subjecting the crystal to simple uniaxial tension. While the stress
strain curves may be plotted in terms of average uniaxial stress vs. aver
age linear strain (AL/Lo), a more fundamental way of presenting the data
is to plot resolved shear stress [Eq. (47)] against the shear strain or glide
strain. Glide strain is the relative displacement of two parallel slip planes
separated at a unit distance. If the orientation of the slip plane and the
slip direction with respect to the tensile axis are known both before and
after deformation, the glide strain y can be obtained^ from Eq. (48)
cos Xi cos Xo . . Qx
7 = . . (48)
sm xi sm Xo
where xo and xi are the initial and final angles between the slip plane and
the tensile axis and Xo and Xi are the initial and final angles between the
slip direction and the tensile axis. The glide strain can also be expressed
in terms of the axial change in length and the original orientation, with
out requiring information on the final orientation of the glide elements.
1^  (1 + 27 sin xo cos Xo + 7 sin^ xo)'^'^ (49)
t>0
or T =
sm
2
— sin^ Xo} — cos \o
(410)
1 S. S. Brenner, /. AppL Phtjs., vol. 27, pp. 14841491, 1956.
2 For a derivation of Eqs. (48) and (49), see E. Schmid and W. Boas, "PlasticiU
of Crystals," English translation, pp. 5860, F. A. Hughes & Co., London, 1950.
Sec. 48j
Plastic Deformation of Single Crystals 103
In the ordinary tension test, the movement of the crosshead of the
testing machine constrains the specimen at the grips, since the grips must
remain in line. Therefore, the specimen is not allowed to deform freely
by uniform glide on every slip plane along the length of the specimen,
as is pictured in Fig. 42 la. Instead, the specimen deforms in the
manner shown in Fig. 4216. Near the center of the gage length the
sV
(a) [b]
Fig. 421. (a) Tensile deformation of
single crystal without constraint; (6)
rotation of slip planes due to con
straint.
Fig. 422. Stereographic triangles show
ing lattice rotation of fee metal during
tensile elongation.
slip planes rotate, as the crystal is extended, so as to align themselves
parallel with the tensile axis. Near the grips bending of the slip planes
is superimposed on the rotation. The amount of rotation toward the
tensile axis increases with the extent of deformation. In tensile defor
mation, the change in the angle between the slip plane and the tensile
axis is related to the change in gage length in the axial direction by
sm xo
sin XI
(411)
A convenient way of recording this reorientation is by following the
axis of the specimen on the unit stereographic triangle.^ In Fig. 422,
the initial orientation of the axis of an fee singlecrystal tension speci
men is plotted on the unit stereographic triangle at P. The slip plane is
1 For a description of stereographic projection, see C. S. Barrett, "The Structure of
Metals," 2d ed., chap. 2, McGrawHill Book Company, Inc., New York, 1952.
104 Metallurgical Fundamentals [Chap. 4
(111), and the slip direction is [101]. During elongation of the crystal,
the specimen axis moves along a great circle passing through P and the
slip direction [101]. As the deformation continues and rotation of the
initial or primarij slip system occurs, the value of cos 4> cos X for the pri
mary slip system decreases. Therefore, even if strain hardening is neg
lected, a greater tensile load must be applied to maintain the value of the
resolved shear stress on this slip system. While cos <^ cos X is decreasing
on the primary slip system owing to rotation, it is increasing on another
set of planes, which are being rotated closer to a position 45° to the tensile
axis. When the resolved shear stress on the new slip system is equal or
about equal to the shear stress on the old slip system, a new set of slip
lines appear on the specimen surface and the axis rotates toward the [112].
In the fee metals, the new slip lines occur on the conjugate slip system
(111)[011]. Under the microscope conjugate slip appears as another set
of intersecting slip lines. Cross slip on the (111)[101] system may also
occur. This slip system has the same slip direction as the primary slip
system. In the microscope cross slip usually appears as short offsets to
the primary slip lines. With even greater rotation, it is geometrically
possible for a fourth slip system (Ill)[011] to become operative. How
ever, this slip system is usually not found to be operative in fee metals.
The appearance of more than one slip system during deformation is often
described under the general heading of duplex or multiple slip.
An excellent method of studying the deformation behavior of single
crystals is by loading in shear. Parker and Washburn ^ have described
a method of loading single crystals in pure shear so that the shear strain
is applied by a couple acting parallel to the active slip system. This
method of testing has the advantage that the crystal can be oriented
so that the maximum shear stress occurs on any desired slip system.
Resolved shear stress and shear strain are measured directly in this type
of test.
49. DeFormation by Twinning
The second important mechanism by which metals deform is the
process known as twinning.^ Twinning results when a portion of the
crystal takes up an orientation that is related to the orientation of the
rest of the untwinned lattice in a definite, symmetrical way. The
twinned portion of the crystal is a mirror image of the parent crystal.
1 E. R. Parker and J. Washburn, "Modern Research Techniques in Physical Metal
lurgy," American Society for Metals, Metals Park, Ohio, 1953.
2 For a complete review of this subject, see E. O. Hall, "Twinning and Diffusionless
Transformations in Metals," Butterworth & Co. (Publishers), Ltd., London, 1954, or
R. W. Cahn, Adv. in Phys., vol. 3, pp. 363445, 1954.
Sec. 49]
Plastic Deformation of Single Crystals
105
The plane of symmetry between the two portions is called the twinning
plane. Figure 423 illustrates the classical atomic picture of twinning.
Figure 423a represents a section perpendicular to the surface in a cubic
lattice with a lowindex plane parallel to the paper and oriented at an
angle to the plane of polish. The twinning plane is perpendicular to the
paper. If a shear stress is applied, the crystal will twin about the twin
ning plane (Fig. 4236). The region to the right of the twinning plane is
undeformed. To the left of this plane, the planes of atoms have sheared
in such a way as to make the lattice a mirror image across the twin plane.
Polished surface
Fig. 423. Classical picture of twinning.
In a simple lattice such as this, each atom in the twinned region moves
by a homogeneous shear a distance proportional to its distance from the
twin plane. In Fig. 4236, open circles represent atoms which have not
moved, dashed circles indicate the original positions in the lattice of atoms
which change position, and solid circles are the final positions of these
atoms in the twinned region. Note that the twin is visible on the
polished surface because of the change in elevation produced by the
deformation and because of the difference in crystallographic orientation
between the deformed and undeformed regions. If the surface were
polished down to section A A, the difference in elevation would be elimi
nated but the twin would still be visible after etching because it possesses
a different orientation from the untwinned region.
It should be noted that twinning differs from slip in several specific
respects. In slip, the orientation of the crystal above and below the slip
plane is the same after deformation as before, while twinning results in
an orientation difference across the twin plane. Slip is usually considered
to occur in discrete multiples of the atomic spacing, while in twinning
the atom movements are much less than an atomic distance. Slip occurs
106 Metallurgical Fundamentals
[Ch
ap.
on relatively widely spread planes, but in the twinned region of a crystal
every atomic plane is involved in the deformation.
Twins may be produced by mechanical deformation or as the result
of annealing following plastic deformation. The first type are known as
mechanical twins; the latter are called annealing twins. Mechanical twins
are produced in bcc or hep metals under conditions of rapid rate of load
ing (shock loading) and decreased temperature. Facecentered cubic
metals are not ordinarily considered to deform by mechanical twinning,
although goldsilver alloys twin fairly readily when deformed at low tem
perature, and mechanical twins have been produced in copper by tensile
deformation at 4°K. Twins can form in a time as short as a few micro
seconds, while for slip there is a delay time of several milliseconds before
a slip band is formed. Under certain conditions, twins can be heard to
form with a click or loud report (tin cry) . If twinning occurs during a
tensile test, it produces serrations in the the stressstrain curve.
Twinning occurs in a definite direction on a specific crystallographic
plane for each crystal structure. Table 45 lists the common twin planes
Twii
Table 45
\i Planes and Twin
Directions
Crystal
structure
Typical
examples
Twin
plane
Twin
direction
bcc
hep
fee
aFe, Ta
Zn, Cd, Mg, Ti
Ag, Au, Cu
(112)
(1012)
(111)
[111]
[1011]
[112]
and twin directions. It is not known whether or not there is a critical
resolved shear stress for twinning. The shear stress at which twinning
occurs is influenced by prior deformation. Figure 420 shows data for
iron single crystals pulled in tension at — 196°C. The solid circles show
a resolved shear stress appreciably below that for slip. The crosses repre
sent iron crystals prestrained 4 per cent at room temperature before test
ing at — 196°C. Note that the resolved shear stress for twinning is
increased by the prior deformation by slip. If the crystals are given
even greater prestrain at room temperature (open circles), deformation
by twinning is completely suppressed and the crystal deforms by slip at
196°C.
The lattice strains needed to produce a twin configuration in a crystal
are small, so that the amount of gross deformation that can be produced
by twinning is small. For example,^ the maximum extension which it is
possible to produce in a zinc crystal when the entire crystal is converted
' Barrett, op. cit., p. 384.
Sec. 49] Plastic DcFormation of Single Crystals 107
into a twin on the {1012} plane is only 7.39 per cent. The important
role of twinning in plastic deformation comes not from the strain pro
duced by the twinning process but from the fact that orientation changes
resulting from twinning may place new slip systems in a favorable orien
tation with respect to the stress axis so that additional slip can take place.
Thus, twinning is important in the overall deformation of metals with a
low number of slip systems, such as the hep metals. However, it should
be understood that only a relatively small fraction of the total volume of
f'"
_^^
4
~y ,

^.^
•y—/
■* /
/
'x^ .,
Fig. 424. Microstructures of twins, (a) Neumann bands in iron; (b) mechanical
twins produced in zinc by polishing; (c) annealing twins in goldsilver alloy.
a crystal is reoriented by twinning, and therefore hep metals will, in
general, possess less ductility than metals with a greater number of slip
systems.
Figure 424 shows some metallographic features of twins in several
different systems. Figure 424a is an example of mechanical twins in
iron (Neumann bands). Note that the width of the twins can be readily
resolved at rather low magnification. The boundaries of the twins etch
at about the same rate as grain boundaries, indicating that they are
rather highenergy boundaries. Figure 4246 shows the broad, lense
shaped twins commonly found in hep metals. Note that twins do not
extend beyond a grain boundary. Figure 424c shows annealing twins
in an fee goldsilver alloy. Annealing twins are usually broader and with
straighter sides than mechanical twins. The energy of annealing twin
boundaries is about 5 per cent of the average grainboundary energy.
Most fee metals form annealing twins. Their presence in the micro
structure is a good indication that the metal has been given mechanical
deformation prior to annealing, since it is likely that they grow fron)
twin nuclei produced during deformation.
108 Metallurgical Fundamentals [Chap. 4
410. Stacking Faults
In an earlier section, it was shown that the atomic arrangement on the
{111 } plane of an fee structure and the {0001 } plane of an hep structure
could be obtained by the stacking of closepacked planes of spheres. For
the fee structure, the stacking sequence of the planes of atoms is given by
ABC ABC ABC. For the hep structure, the stacking sequence is given
by AB AB AB.
^o o^ So off
^o ~^C^ ^° Z^'^
Bq Op Cq^ o
tOA
Be
o
A B C A BCA A B C A\C A B
(a) {b]
TO
Ar.^
'o— o° o OB
^o oc o o A
^^^ ^o °5
^o o a o A
o o o OB
o o^ o A
A BC'A CB'CA ABABAB
I I
(c) {d)
Fig. 425. Faulted structures, (a) Facecentered cubic packing; (6) deformation
fault in fee; (c) twin fault in fee; id) hep packing.
Fairly recently it has been realized that errors, or faults, in the stack
ing sequence can be produced in most metals by plastic deformation.^
Slip on the {111} plane in an fee lattice produces a deformation stacking
fault by the process shown in Fig. 4255. Slip has occurred between
an A and a B layer, moving each atom layer above the slip plane one
identity distance to the right. The stacking sequence then becomes
ABC A CAB. Comparison of this faulted stacking sequence (Fig.
4256) with the stacking sequence for an hep structure without faults
(Fig. 425d) shows that the deformation stacking fault contains four
layers of an hep sequence. Therefore, the formation of a stacking fault
^ Very precise Xray diffraction measurements are needed to detect the presence of
stacking faults. For example, see B. E. Warren and E. P. Warekois, Acta Met.,
vol. 3, p. 473, 1955.
Sec. 410] Plastic Deformation of Single Crystals 109
in an fee metal is equivalent to the formation of a thin hep region.
Another way in which a stacking fault eould oeeur in an fee metal is by
the sequence' shown in Fig. 425c. The stacking sequence ABC\ACB\CA
is called an extrinsic, or twin, stacking fault. The three layers ACB
constitute the twin. Thus, stacking faults in fee metals can also be con
sidered as siibmicroscopic twins ot nearly atomic thickness. The reason
why mechanical twins of microscopically resolvable width are not formed
readily when fee metals are deformed is that the formation of stacking
faults is so energetically favorable.
The situation for the hep structure is somewhat different from that
found in fee metals. Figure 425(i shows that, on going from an A layer
to a B layer, if we continue in a
straight Hne we will not come stacLing f^uit^ ^siip plane
^ Partiol ^ '^
to another atom on the next A disiocationsr^^y ) /
layer. However, slip can occur / /^\ J
between tw^o of the planes so that //r: ■■:;■.;. ■:■.■•^^J^>>^
the stacking seciuence becomes i^sA^.\>.^_>.i' ^<ui^
ABABACBCBC. As a result, Z /
four layers of atoms BACB are Fig. 426. Schematic model of a stacking
in the straightline fee stacking fault.
order. Thus, a stacking fault in
an hep metal is equivalent to the formation of a thin fee region. It is
more difficult to form stacking faults in a bee lattice than in the close
packed fee and hep structures. The possibility of stacking faults in the
{112} planes has been investigated theoretically and demonstrated by
Xray diffraction." Stacking faults have been observed w^ith thinfilm
electron microscopy in columbium.^
Stacking faults occur most readily in fee metals, and they have been
most extensively studied for this crystal structure. For example, it is
now known that differences in the deformation behavior of fee metals
can be related to differences in staekingfault behavior. From the point
of view of dislocation theory, a stacking fault in an fee metal can be con
sidered to be an extended dislocation consisting of a thin hexagonal region
bounded by partial dislocations^ (Fig. 426). The nearly parallel dis
locations tend to repel each other, but this is balanced by the surface
tension of the stacking fault pulling them together. The lower the
stacking fault energy, the greater the separation between the partial dis
1 C. N. J. Wagner, Acta Met., vol. 5, pp. 427434. 1957.
2 P. B. Hirsch and H. M. Otte, Acta Cryst., vol. 10, pp. 44745.3, 1957; O. J. Guenter
and B. E. Warren, /. Appl. Phtjs., vol. 29, pp. 4048, 1958.
3 A. Fourdeux and A. Berghezen, J. Inst. Metals, vol. 89, pp. 3132, 19601961.
* Partial dislocations will be considered in more detail in Chap. 6. The splitting
of dislocations into separated partials has been observed with the electron microscope
in stainlesssteel foils.
110 Metallurgical Fundamentals [Chap. 4
locations and the wider the stacking fault. Stackingfault energies in
fee metals have been estimated on the assumption that the stacking
fault energy is equal to twice the energy of a coherent boundary of an
annealing twin. On this basis, the stackingfault energies for copper,
nickel, and aluminum are approximately 40, 80, and 200 ergs/cm^. Since
the lower the energy of the twin boundary, the greater the tendency for
the formation of annealing twins, the estimates of stackingfault energy
are in qualitative agreement with metallographic observations of the fre
quency of occurrence of annealing twins; e.g., aluminum rarely shows
annealing twins. Xray work has shown that the energy of stacking
faults in brass decreases with zinc content, and this is in agreement with
the fact that alpha brass forms a greater number of annealing twins than
copper.
Stacking faults enter into the plastic deformation of metals in a number
of ways. Metals with wide stacking faults strainharden more rapidly,
twin easily on annealing, and show a different temperature dependence
of flow stress from metals with narrow stacking faults. Figure 426 helps
to illustrate why cross slip is more difficult in metals with wide stacking
fault ribbons. Because dislocations in the slip plane are extended, it is
not possible for them to transfer from one slip plane to another except
at a point where the partial dislocations come together. Since it requires
energy to produce a constriction in the stacking fault, the process of cross
slip is more difficult in a metal with wide stacking faults than in a metal
with narrow stacking faults. For example, the activation energy for
cross slip is about 1 ev in aluminum and approximately 10 ev in copper.
411. DeFormation Bands and Kink Bands
Inhomogeneous deformation of a crystal results in regions of different
orientation called deformation bands. When slip occurs without restraint
in a perfectly homogeneous fashion, the slip lines are removed by subse
quent polishing of the surface. Deformation bands, however, can be
observed even after repeated polishing and etching because they repre
sent regions of different crystallographic orientation. In single crystals,
deformation bands several millimeters wide may occur, while in poly
crystalline specimens microscopic observation is needed to see them.
The tendency for the formation of deformation bands is greater in poly
crystalline specimens because the restraints imposed by the grain bound
aries make it easy for orientation differences to arise in a grain during
deformation. Deformation bands generally appear irregular in shape
but are elongated in the direction of principal strain. The outline of
the bands is generally indistinct and poorly defined, indicating a general
Sec. 412] Plastic Deformation of Sinsle Crystals 111
fading out of the orientation difference. Deformation bands have been
observed in both fee and bee metals, but not in hep metals.
Consideration of the equation for critical resolved shear stress shows
that it will be difficult to deform a hexagonal crystal when the basal plane
is nearly parallel to the crystal axis. Orowan^ found that if a cadmium
crystal of this orientation w^re
loaded in compression it would
deform by a localized region of the
crystal suddenly snapping into a
tilted position with a sudden short
ening of the crystal. The buckling,
or kinking, behavior is illustrated
in Fig. 427. The horizontal lines
represent basal planes, and the
Fig. 427. Kink band.
planes designated p are the kink planes at which the orientation sud
denly changes. Distortion of the crystal is essentially confined to the
kink band. Further study of kink bands by Hess and Barrett^ showed
that they can be considered to be a simple type of deformation band.
Kink bands have also been observed in zinc crystals tested in tension,
where a nonuniform distribution of slip can produce a bending moment
which can cause kink formation.
412. Strain Hardenins oF Sinsle Crystals
One of the chief characteristics of the plastic deformation of metals
is the fact that the shear stress required to produce slip continuously
increases with increasing shear strain. The increase in the stress required
to cause slip because of previous plastic deformation is known as strain
hardening, or work hardening. An increase in flow stress of over 100 per
cent from strain hardening is not unusual in single crystals of ductile
metals.
Strain hardening is caused by dislocations interacting with each other
and with barriers which impede their motion through the crystal lattice.
Hardening due to dislocation interaction is a complicated problem because
it involves large groups of dislocations, and it is difficult to specify group
behavior in a simple mathematical way. It is known that the number
of dislocations in a crystal increases with strain over the number present
in the annealed crystal. Thus, the first requirement for understanding
strain hardening was the development of a logical mechanism for the
generation of dislocations. F. C. Frank and W. T. Read conceived a
1 E. Orowan, Nature, vol. 149, p. 643, 1942.
2 J. A. Hess and C. S. Barrett, Trans. AIME, vol. 185, p. 599, 1949.
US Metallutgical Fundamentafs
[Qap. 4
Slip direction J 80 ° to
original direction ,
logical mechanism by which a large amount of shp couJd be produced
by one dislocation. The FrankRead source (see Chap. 6 for details)
provides a method by which the dislocations initially present in the crystal
as a result of growth can generate enough dislocations to account for the
observed strain hardening. The mechanism is consistent with the experi
mental observation that slip is concentrated on a relatively few active
slip planes and that the total slip on each slip plane is of the order of
1,000 atomic spacings. A method is also provided in the concept of the
FrankRead source for immobilizing the source after slip of this order of
magnitude has occurred. Direct ex
perimental evidence for the existence
of the FrankRead source in crystals
has been developed in recent years.
One of the earliest dislocation con
cepts to explain strain hardening was
the idea that dislocations pile up on
slip planes at barriers in the crystal.
The pileups produce a hack stress
which opposes the applied stress on
the slip plane. The existence of a
back stress was demonstrated experi
mentally by shear tests on zinc single
crystals.^ Zinc crystals are ideal for
Shear strain y
Fig. 428. Effect of complete reversal of
slip direction on stressstrain curve.
(E. H. Edwards, J. Washburn, and E. R.
Parker, Trans. AIMS, vol., 197, p. 1526,
1953.)
crystalplasticity experiments because they slip only on the basal plane,
and hence complications due to duplex slip are easily avoided. In Fig.
428, the crystal is strained to point 0, unloaded, and then reloaded in
the direction opposite to the original slip direction. Note that on reload
ing the crystal yields at a lower shear stress than when it was first loaded.
This is because the back stress developed as a result of dislocations piling
up at barriers during the first loading cycle is aiding dislocation move
ment when the direction of slip is reversed. Furthermore, when the slip
direction is reversed, dislocations of opposite sign could be created at the
same sources that produced the dislocations responsible for strain in the
first slip direction. Since dislocations of opposite sign attract and annihi
late each other, the net effect would be a further softening of the lattice.
This explains the fact that the flow curve in the reverse direction lies
below the curve for continued flow in the original direction. The lower
ing of the yield stress when deformation in one direction is followed by
deformation in the opposite direction is called the Bauschinger effect.^
While all metals exhibit a Bauschinger effect, it may not always be of
1 E. H. Edwards, J. Washburn, and E. R. Parker, Trans. AIME, vol. 197, p. 1525,
1953.
2 J. Bauschinger, Zivilingur., vol. 27, pp. 289347, 1881.
Sec. 41 2]
Plastic Deformation of Single Crysta!
113
the magnitude shown here for zinc crystals. Moreover, the flow curve
after reversal of direction does not fall below the original flow curve for
all metals.
The existence of back stress and its importance to strain hardening in
metals having been established, the next step is to identify the barriers
to dislocation motion in single crystals. Microscopic precipitate parti
cles and foreign atoms can serve as barriers, but other barriers which are
effective in pure single crystals must be found. Such barriers arise from
the fact that glide dislocations on intersecting slip planes may combine
Fig. 429. Schematic representation of intersection of two screw dislocations, (a)
Before intersection; (6) jogs formed after intersection.
with one another to produce a new dislocation that is not in a slip direc
tion. The dislocation of low mobility that is produced by a dislocation
reaction is called a sessile dislocation. Since sessile dislocations do not lie
on the slip plane of low shear stress, they act as a barrier to dislocation
motion until the stress is increased to a high enough level to break down
the barrier. The most important dislocation reaction, which leads to
the formation of sessile dislocations, is the formation of CottrellLomer
barriers in fee metals by slip on intersecting {111} planes.
Another mechanism of strain hardening, in addition to that due to the
back stress resulting from dislocation pileups at barriers, is believed to
occur when dislocations moving in the slip plane cut through other dis
locations intersecting the active slip plane. The dislocations threading
through the active slip plane are often called a dislocation forest, and this
strainhardening process is referred to as the intersection of a forest of
dislocations. Figure 429 shows that the intersection of dislocations
results in the formation of jogs, or offsets, in the dislocation line. The
jogs formed in this case are edge dislocations because their Burgers vec
tors are perpendicular to the original dislocation line. Any further move
ment of the screw dislocations along the line A A would require the newly
formed edge components to move out of their slip planes. Thus, the
114 Metallurgical Fundamentals
[Ch
ap.
formation of jogs in screw dislocations impedes their motion and may
even lead to the formation of vacancies and interstitials if the jogs are
forced to move nonconservatively. Jogs in edge dislocations do not
impede their motion. All these processes require an increased expendi
ture of energy, and therefore they contribute to hardening.
Strain hardening due to a dislocation cutting process arises from short
range forces occurring over distances less than 5 to 10 interatomic dis
tances. This hardening can be overcome at finite temperatures with the
^2 ^3
Resolved shear strain y
Fig. 430. Generalized flow curve for fee single crystals.
help of thermal fluctuations, and therefore it is temperature and strain
ratedependent. On the other hand, strain hardening arising from dis
location pileup at barriers occurs over longer distances, and therefore it
is relatively independent of temperature and strain rate. Accordingly,
data on the temperature and strainrate dependence of strain hardening
can be used^ to determine the relative contribution of the two mechanisms.
When the stressstrain curves for single crystals are plotted as resolved
shear stress vs. shear strain, certain generalizations can be made for all
fee metals. Following the notation proposed by Seeger,^ the flow curve
for puremetal single crystals can be divided into three stages (Fig. 430) .
Stage I, the region of easy glide, is a stage in which the crystal undergoes
little strain hardening. During easy glide, the dislocations are able to
move over relatively large distances without encountering barriers. The
low strain hardening produced during this stage implies that most of the
dislocations escape from the crystal at the surface. During easy glide,
1 Z. S. Basinski, Phil. Mag., vol. 4, ser. 8, pp. 393432, 1959.
2 A. Seeger, "Dislocations and Mechanical Properties of Crystals,'
Sons, Inc., New York, 1957.
John Wiley &
Sec. 412] Plastic Deformation oF Single Crystals 115
slip always occurs on only one slip system. For this reason, stage I slip
is sometimes called laminar flow.
Stage II is a nearly linear part of the flow curve where strain hardening
increases r.apidly. In this stage, slip occurs on more than one set of
planes. The length of the active slip lines decreases with increasing
strain, which is consistent with the formation of a greater number of
CottrellLomer barriers with increasing strain. During stage II, the
ratio of the strainhardening coefficient (the slope of the curve) to the
shear modulus is nearly independent of stress and temperature, and
approximately independent of crystal orientation and purity. The fact
that the slope of the flow curve in stage II is nearly independent of tem
perature agrees with the theory that assumes the chief strainhardening
mechanism to be piledup groups of dislocations.
Stage III is a region of decreasing rate of strain hardening. The proc
esses occurring during this stage are often called dynamical recovery. In
this region of the flow curve, the stresses are high enough so that dis
locations can take part in processes that are suppressed at lower stresses.
Cross slip is believed to be the main process by which dislocations, piled
up at obstacles during stage II, can escape and reduce the internalstrain
field. The stress at which stage III begins, T3, is strongly temperature
dependent. Also, the flow stress of a crystal strained into stage III is
more temperaturedependent than if it had been strained only into stage
II. This temperature dependence suggests that the intersection of forests
of dislocations is the chief strainhardening mechanism in stage III.
The curve shown in Fig. 430 represents a general behavior for fee
metals. Certain deviations from a threestage flow curve have been
observed. For example, metals with a high stackingfault energy, like
aluminum, usually show only a very small stage II region at room tem
perature because they can deform so easily by cross slip. The shape and
magnitude of a singlecrystal flow curve, particularly during the early
stages, depends upon the purity of the metal, the orientation of the
crystal, the temperature at which it is tested, and the rate at which it is
strained. The easyglide region is much more prominent in hep crystals
than in fee metals. A region of easy glide in the flow curve is favored by
slip on a single system, high purity, low temperature, absence of surface
oxide films, an orientation favorable for simple slip, and a method of
testing which minimizes extraneous bending stresses. Figure 431 shows
that crystal orientation can have a very strong eff"ect on the flow curve
of fee single crystals. When the tensile axis is parallel to a (Oil) direc
tion, one slip system is carrying appreciably more shear stress than any
other and the flow curve shows a relatively large region of easy glide.
When the tensile axis is close to a (100) or (111) direction, the stress on
several slip systems is not very different and the flow curves show rapid
rates of strain hardening.
116 Metallurgical Fundamentals
[Ch
ap.
Starting as close to absolute zero as is practical, the value of the
resolved shear stress at a given shear strain decreases with increasing
temperature. If fee crystals are strained to the end of stage II at a tem
perature Ti and then the temperature is increased to T2 without any
Resolved shear strain
Fig. 431. Effect of specimen orientation on the shape of the flow curve for fee single
crystals.
Shear strain
Fig. 432. Flow curves exhibiting work softening.
change in strain, the flow stress drops from ri to t2 (Fig. 432). The
state of strain hardening reached at Ti is unstable at T2, and a recovery
process sets in which tends to reduce the strain hardening to what it
would have been if all the straining had been accomplished at T2. This
behavior is called^ work softening. Work softening is the result of the
release at T2 of dislocation pileups produced at Ti. The release of dis
locations may be due to easier cross shp at the higher temperature or
1 A. H. Cottrell and R. J. Stokes, Proc. Roy. Soc. (London), vol. A233, p. 17, 1955.
Sec. 412] Plastic Deformation of Sinsle Crystals 117
the fact that the size of a stable dislocation pileup is smaller at T2 because
of increased thermal fluctuations.
BIBLIOGRAPHY
Azaroff, L. V.: "Introduction to Solids," McGrawHill Book Company, Inc., New
York, 1960.
Barrett, C. S.: "The Structure of Metals," 2d ed., McGrawHill Book Company,
Inc., New York, 1952.
Clarebrough, L. M., and M. E. Hargreaves: Work Hardening of Metals, in "Progress
in Metal Physics," vol. 8, Pergamon Press, Ltd., London, 1959.
Cottrell, A. H.: "Dislocations and Plastic Flow in Crystals," Oxford University Press,
New York, 1953.
Maddin, R., and N. K. Chen: Geometrical Aspects of the Plastic Deformation of
Metal Single Crystals, in "Progress in Metal Physics," vol. 5, Pergamon Press,
Ltd., London, 1954.
Schmid, E., and W. Boas: "Plasticity of Crystals," English translation, F. A. Hughes
& Co., London, 1950.
Chapter 5
PLASTIC DEFORMATION
OF POLYCRYSTALLINE AGGREGATES
51 . Introduction
The previous chapter considered the plastic deformation of metallic
single crystals in terms of the movement of dislocations and the basic
deformation mechanisms of slip and twinning. Singlecrystal specimens
represent the metal in its most ideal condition. The simplification which
results from the singlecrystal condition materially assists in describing
the deformation behavior in terms of crystallography and defect struc
ture. However, with the exception of electronic and semiconductor
devices, single crystals are rarely used for practical purposes because of
limitations involving their strength, size, and production. Commercial
metal products are invariably made up of a tremendous number of small
individual crystals or grains. The individual grains of the polycrystalline
aggregate do not deform in accordance with the relatively simple laws
which describe plastic deformation in single crystals because of the
restraining effect of the surrounding grains. Therefore, there is a gap
between fundamental deformation mechanisms determined from single
crystals and the prediction of the plastic behavior of a polycrystalline
aggregate from these basic concepts.
Grain boundaries exert a considerable influence on the plasticdeforma
tion behavior of polycrystalline metals. Other factors which also have
an important effect on mechanical properties are the presence of sub
grain boundaries within the grains, solidsolution alloying additions,
and dispersion of secondphase particles. These factors will each
be considered in this chapter, primarily in terms of how they
influence the tensileflow curve. Wherever possible, qualitative explana
tions of these processes will be given in terms of dislocation theory.
Other topics covered in this chapter include yieldpoint behavior, strain
aging, cold work, annealing, and the development of preferred orienta
tions. It will be appreciated that not all these topics are solely restricted
to polycrystalline materials. However, the bulk of the experimental
118
Sec. 52] Plastic Deformation oF Polycrystalline Aggregates 119
data on these phenomena have been obtained from polycrystalHne
materials, and therefore they are considered in this chapter.
52. Grain Boundaries and Deformation
The boundaries between grains in a polycrystalline aggregate are a
region of disturbed lattice only a few atomic diameters wide. In the
general case, the crystallographic orientation changes abruptly in passing
from one grain to the next across the grain boundary. The ordinary
highangle grain boundary represents a region of random misfit between
the adjoining crystal lattices.^ As the difference in orientation between
the grains on each side of the boundary decreases, the state of order in the
boundary increases. For the limiting case of a lowangle boundary
where the orientation difference across the boundary may be less than 1°
(see Sec. 53), the boundary is composed of a regular array of dislocations.
Ordinary highangle grain boundaries are boundaries of rather high
surface energy. For example, a grain boundary in copper has an inter
facial surface energy of about 600 ergs/cm^, while the energy of a twin
boundary is only about 25 ergs/cm^. Because of their high energy, grain
boundaries serve as preferential sites for solidstate reactions such as
diffusion, phase transformations, and precipitation reactions. An
important point to consider is that the high energy of a grain boundary
usually results in a higher concentration of solute atoms at the boundary
than in the interior of the grain. This makes it difficult to separate the
pure mechanical effect of grain boundaries on properties from an effect
due to impurity segregation.
Grain boundaries may serve to either strengthen or weaken a metal,
depending upon the temperature, rate of strain, and the purity of the
metal. At temperatures below approximately onehalf of the absolute
melting point, and for relatively fast strain rates (so that recovery effects
are not great), grain boundaries increase the rate of strain hardening and
increase the strength. At high temperatures and slow strain rates (con
ditions of creep deformation) deformation is localized at the grain
boundaries. Grainboundary sliding and stressinduced migration can
occur, and eventually fracture takes place at the grain boundary. The
fairly narrow temperature region in which the grain boundaries become
weaker than the interior of the grains, so that fracture occurs in an inter
granular rather than transgranular fashion, is called the equicohesive
temperature.
The principal difference between the roomtemperature deformation of
singlecrystal and polycrystalline specimens is that polycrystalline
1 For a review of the proposed models of grain boundaries see D. McLean, "Grain
Boundaries in Metals," chap. 2, Oxford University Press, New York 19.57
120 Metallurgical Fundamentals [Chap. 5
material exhibits a higher rate of strain hardening. The stressstrain
curve for polycrystaUine material shows no stage I or easyglide region.
Only stage II and stage III deformation are obtained with polycrystaUine
specimens. Associated with the increased strain hardening is usually an
increase in yield stress and tensile strength. The effects of grain bound
aries on strength are due to two main factors. The first is the fact that
Fig. 51 . Dislocations piled up against a grain boundary, as observed with the electron
microscope in a thin foil of stainless steel, 17,500 X. [M. J. Whelan, P. B. Hirsch,
R. W. Home, and W. Bollman, Proc. Roy. Soc. (London), vol. 240A, p. 524, 1957.]
grain boundaries are barriers to slip. Of greater importance is the fact
that the requirement for continuity between grains during deformation
introduces complex modes of deformation within the individual grains.
Slip on multipleslip systems occurs very readily in polycrystaUine
specimens.
The fact that slip lines stop at grain boundaries can be readily observed
with the light microscope. However, by means of special etchpit
techniques (Sec. 62) and highmagnification electron microscopy of thin
films it is possible to establish that dislocations pile up along the slip
planes at the grain boundaries (Fig. 51). Dislocation pileups produce
back stresses which oppose the generation of new dislocations at Frank
Read sources within the grains. With increasing applied stress more and
more dislocations pile up at grain boundaries. High shear stresses are
developed at the head of a dislocation pileup, and eventually this becomes
high enough to produce dislocation movement in the neighboring grain
across the boundary. This will reduce the dislocation pileup and mini
mize hardening from this effect. Hardening due to dislocation pileup
Sec. 521 Plastic Deformation of Polycrystalline Aggregates 121
at grain boundaries is therefore important in the early stages of deforma
tion, but not at large strains. It will be more effective in an hep metal,
with only one easy slip plane, than in fee or bee metals, with many
equivalent slip planes. For the latter case, no grain can be very unfavor
ably oriented with respect to the applied stress, so that, on the average,
slip can be initiated in a neighboring grain at only a little higher stress
than was required to initiate slip in the most favorably oriented grains.
However, for hep metals, there may be a very unfavorable orientation
difference between neighboring grains so that an appreciably higher stress
is required to initiate slip in the neighboring grain. Therefore, poly
crystalline hep metals show a very much higher rate of strain hardening
compared with single crystals. In fee and bcc metals the difference in the
flow curve between polycrystals and single crystals is not nearly so great.
The effect of crystal orientation on the flow curve of fee single crystals
was illustrated in Fig. 431. Orientations which produce many favorably
oriented slip systems readily deform by multiple slip. Multiple slip
always results in a high rate of strain hardening. From purely geometri
cal considerations the grains of a polycrystalline metal must remain in
contact during deformation. Taylor^ has shown that five independent
slip systems must operate in each grain in order to maintain continuity.
Since slip on only two or three systems, depending on orientation, occurs
for multiple slip in single crystals, slip in polycrystals is more complex
than in single crystals oriented for multiple slip. Greater strain harden
ing is usually observed in polycrystals than can be accounted for on the
basis of multiple slip in single crystals and by grainboundary barriers.^
Grain size has a measurable effect on most mechanical properties. For
example, at room temperature, hardness, yield strength, tensile strength,
fatigue strength, and impact resistance all increase with decreasing grain
size. The effect of grain size is largest on properties which are related to
the early stages of deformation, for it is at this stage that grainboundary
barriers are most effective. Thus, yield stress is more dependent on
grain size than tensile strength. For the later stages of deformation the
strength is controlled chiefly by complex dislocation interactions occurring
within the grains, and grain size is not a controlling variable.
For most metals the yield stress is related to the grain size by
cro = ai + K,Dy^ (51)
where an = yield stress
(Xi = friction stress opposing motion of a dislocation
K,, = measure of extent to which dislocations are piled up at
barriers
D = grain diameter
1 G. I. Taylor, J. Inst. Metals, vol. 62, p. 307, 1938.
2 McLean, op. cit., chap. 6.
122 Metallurgical Fundamentals [Chap. 5
Equation (51) was first proposed for lowcarbon steeP and has been
extensively applied to tests on this material. The slope of a plot of oq
versus D~^^ is Ky, a measure of the extent to which dislocations are piled
up at grain boundaries. It is essentially independent of temperature.
The intercept ai is a measure of the stress needed to drive a dislocation
against the resistance of impurities, precipitate particles, subgrain bound
aries, and the PeierlsNabarro force. This term depends on both the
composition and the temperature, but it is independent of the applied
stress. Since the PeierlsNabarro force is temperaturedependent and
the other resistances to dislocation motion are approximately tempera
tureindependent, it appears possible to obtain an estimate of the lattice
resistance to dislocation motion from an analysis of the grainsize depend
ence of yield stress."
The problem of determining the flow curve of polycrystalline material
from singlecrystal data is difficult. The analyses of this problem which
have been made^ consist essentially in averaging the singlecrystal curves
over different orientations. Only moderate agreement has been obtained.
Grain size is measured with a microscope by counting the number of
grains within a given area, by determining the number of grains that
intersect a given length of random line, or by comparison with standard
charts. The average grain diameter D can be determined from measure
ments along random lines by the equation
D = ^ (52)
where L is the length of the line and N is the number of intercepts which
the grain boundary makes with the line. This can be related^ to the
ratio of the grainboundary surface area S to the volume of the grains,
V, by the equation
where I is the total length of grain boundary on a random plane of polish
and A is the total area of the grains on a random plane of polish. A very
common method of measuring grain size in the United States is to com
pare the grains at a fixed magnification with the American Society for
Testing Materials (ASTM) grainsize charts. The ASTM grainsize
1 N. J. Fetch, /. Iron Steel Inst. (London), vol. 173, p. 25, 1953; E. O. Hall, Proc.
Phys. Soc. (London), vol. 64B, p. 747, 1951.
2 J. Heslop and ISl J. Fetch, Phil. Mag., vol. 1, p. 866, 1956.
3 Taylor, op. cit.; J. F. W. Bishop, J. Mech. and Phys. Solids, vol. 3, pp. 259266,
1955; U. F. Kocks, Ada Met., vol. 8, pp. 345352, 1960.
^ C. S. Smith and L. Guttman, Trans. AIMS, vol. 197, p. 81, 1953.
Sec. 53]
Plastic Deformation of Polycrystalline Aggregates 123
number n is related to N*, the number of grains per square inch at a
magnification of lOOX by the relationship s ,
N* = 2'
(54)'
Table 51 compares the ASTM grainsize numbers with several other
useful measures of grain size.
Table 51
Comparison of Grainsize Measuring Systemsj
ASTM
No.
Grains/in. 2 at 100 X
Grains /mm 2
Grains /mm. 3
Av. grain
diam, mm
3
0.06
1
0.7
1.00
2
0.12
2
2
0.75
1
0.25
4
5.6
0.50
0.5
8
16
035
1
1
16
45
0.25
2
2
32
128
0.18
3
4
64
360
0.125
4
8
128
1,020
0.091
5
16
256
2,900
0.062
6
32
512
8,200
0.044
7
64
1,024
23,000
G 032
8
128
2,048
65,000
0.022
9
256
4,096
185,000
0.016
10
512
8,200
520,000
0.011
11
1,024
16,400
1,500,000
0.008
12
2,048
32,800
4,200,000
0.006
t ASM Metals Handbook, 1948 ed.
53. Lowangle Grain Boundaries
It has been recognized only fairly recently that a definite substructure
can exist within the grains surrounded by highenergy grain boundaries.
The subgrains are lowangle boundaries in which the difference in orienta
tion across the boundary may be only a few minutes of arc or, at most, a
few degrees. Because of this small orientation difference, special Xray
techniques are required to detect the existence of a substructure network.
Subgrain boundaries are lowerenergy boundaries than grain boundaries,
and therefore they etch less readily than grain boundaries. However,
in many metals they can be detected in the microstructure by metallo
graphic procedures (Fig. 52).
A lowangle boundary contains a relatively simple arrangement of
dislocations. The simplest situation is the case of a tilt boundary.
Figure 53a illustrates two cubic crystals with a common [001] axis. The
124 Metallurgical Fundamentals [Chap. 5
slight difference in orientation between the grains is indicated by the
angle 6. In Fig. 536 the two crystals have been joined to form a bicrystal
containing a lowangle boundary. Along the boundary the atoms adjust
their position by localized deformation to produce a smooth transition
from one grain to the other. However, elastic deformation cannot
accommodate all the misfit, so that some of the atom planes must end on
the grain boundary. Where the atom planes end, there is an edge dis
location. Therefore, lowangle tilt boundaries can be considered to be
A »;
^
Fig. 52. Substructure network in iron3 per cent silicon alloy, 250 X.
an array of edge dislocations. From the geometry of Fig. 536 the
relationship between d and the spacing between dislocations is given by
^ = 2tani2^ = ^ (55)
where h is the magnitude of the Burgers vector of the lattice.
The validity of the dislocation model of the lowangle boundary is
found in the fact it is possible to calculate the grainboundary energy as
a function of the difference in orientation between the two grains. So
long as the angle does not become greater than about 20°, good agreement
is obtained between the measured values of grainboundary energy and
the values calculated on the basis of the dislocation model. Other
evidence for the dislocation nature of lowangle boundaries comes from
metallographic observations. If the angle is low, so that the spacing
between dislocations is large, it is often possible to observe that the
Sec. 53]
Plastic Deformation of Polycrystalline Aggregates 125
ia) id)
Fig. 53. Diagram of lowangle grain boundary, (a) Two grains having a common
[001] axis and angular difference in orientation of 6; (b) two grains joined together to
form a lowangle grain boundary made up of an array of edge dislocations. {W. T.
Read, Jr., ^'Dislocations in Crystals," p. 157, McGrawHill Book Company, Inc., New
York, 1953.)
Fig. 54. Etchpit structures along lowangle grain boundaries in ironsilicon alloy,
1,000 X.
126 Metallurgical Fundamentals
[Chap. 5
boundary is composed of a row of etch pits, with each pit corresponding
to the site of an edge dislocation (Fig. 54).
Subboundaries or lowangle boundaries can be produced in a number
(a) id)
Fig. 55. Movement of dislocations to produce polygonization (schematic).
of ways.^ They may be produced during crystal growth, during high
temperature creep deformation, or as the result of a phase transformation.
The veining in ferrite grains is a wellknown example of a substructure
resulting from the internal stresses ac
companying a phase transformation.
Perhaps the most general method of
producing a substructure network is
by introducing a small amount of de
formation (from about 1 to 10 per
cent prestrain) and following this with
an annealing treatment to rearrange
the dislocations into subgrain bound
aries. The amount of deformation
and temperature must be low enough
to prevent the formation of new grains
by recrystallization (see Sec. 512).
This process has been called recrystal
lization in situ, or polygonization.
The term polygonization was used
originally to describe the situation that
occurs when a single crystal is bent to
a relatively small radius of curvature
and then annealed. Bending results
in the introduction of an excess number of dislocations of one sign.
These dislocations are distributed along the bentglide planes as shown
in Fig. 55a. When the crystal is heated, the dislocations group them
selves into the lowerenergy configuration of a lowangle boundary by
dislocation climb. The resulting structure is a polygonlike network of
lowangle grain boundaries (Fig. 556).
^ R. W. Cahn, "Impurities and Imperfections," American Society for Metals,
Metals Park, Ohio, 1955.
1 2 3 4 5 6
Density of subboundaries
(arbitrary scale)
Fig. 56. Effect of density of sub
boundaries on yield stress. (E. R.
Parker and T. H. Hazlett, "Relation
of Properties to Microstructure,"
American Society for Metals, Metals
Park, Ohio, 1954.)
53]
Plastic Deformation of Polycrystalline Agsregatcs 127
Since lowangle boundaries consist of simple dislocation arrays, a study
of their properties should provide valuable information on dislocation
behavior. Parker and Washburn ^ demonstrated that a lowangle bound
ary moves as a unit when subjected to a shear stress, in complete agree
ment with what would be expected for a linear dislocation array. It has
also been found that the boundary angle decreases with increasing
80
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Aircraft quality AISI C1020 steel
A// specimens annealed initially
I275°F forZhr
• As annealed
^ Coldreduced 8 % in thickness
D Cold reduced 8% annealed
1 27 5° F for V2 hr oilquenched
Strain rote, 0.002 in./in./min
?
0.10 0.20
Engineering strain, in. /in.
0.30
Fig. 57. Effect of a substructure of lowangle grain boundaries on the stressstrain
curve of SAE 1020 steel. {E. R. Parker and J. Washburn, ''Impurities and Imperfec
tions," p. 155, American Society for Metals, Metals Park, Ohio, 1955.)
distance of shear. This means that the boundary loses dislocations as it
moves, a fact which would be expected if dislocations are held up at
imperfections such as foreign atoms, precipitated particles, and other
dislocations.
The formation of subgrains in an annealed material results in a signifi
cant increase in strength. Figure 56 shows the increase in the yield
stress in nickel due to an increase in the density of subgrain boundaries
produced by various prestraining and anneaUng treatments. The fact
that the curves for pure nickel and alloyed nickel are nearly parallel
indicates that the strengthening due to substructure is additive to that
produced by solidsolution hardening. The effect of a substructure of
lowangle grain boundaries on the stressstrain curve of 1020 steel is
shown in Fig. 57. Note that the material that was coldreduced and
1 E. R. Parker and J. Washburn, Trans. AIME, vol. 194, pp. 10761078, 1952.
128 Metallurgical Fundamentals [Chap. 5
annealed, so as to produce a substructure, has a higher yield point and
tensile strength than both the annealed material and the material which
was only coldreduced. Moreover, the ductility of the material contain
ing a substructure is almost as good as the ductility of the annealed steel.
54. Solidsolution Hardening
The introduction of solute atoms into solid solution in the solvent
atom lattice invariably produces an alloy which is stronger than the pure
metal. There are two types of solid solutions. If the solute and solvent
atoms are roughly similar, the solute atoms will occupy lattice points in
the crystal lattice of the solvent atoms. This is called substitutional
solid solution. If the solute atoms are much smaller than the solvent
atoms, they occupy interstitial positions in the solvent lattice. Carbon,
nitrogen, oxygen, hydrogen, and boron are the elements which commonly
form interstitial solid solutions.
The factors which control the tendency for the formation of substitu
tional solid solutions have been uncovered chiefly through the work of
HumeRothery. If the sizes of the two atoms, as approximately indi
cated by the lattice parameter, differ by less than 15 per cent, the size
factor is favorable for solidsolution formation. When the size factor is
greater than 15 per cent, the extent of solid solubility is usually restricted
to less than 1 per cent. Metals which do not have a strong chemical
affinity for each other tend to form solid solutions, while metals which are
far apart on the electromotive series tend to form intermetallic com
pounds. The relative valence of the solute and solvent also is important.
The solubility of a metal with higher valence in a solvent of lower valence
is more extensive than for the reverse situation. For example, zinc is
much more soluble in copper than is copper in zinc. This relativevalence
effect can be rationalized to a certain extent in terms of the electronatom
ratio. ^ For certain solvent metals, the limit of solubility occurs at
approximately the same value of electronatom ratio for solute atoms of
different valence. Finally, for complete solid solubility over the entire
range of composition the solute and solvent atoms must have the same
crystal structure.
The acquisition of fundamental information about the causes of solid
solution hardening has been a slow process. Early studies'' of the increase
in hardness resulting from solidsolution additions showed that the hard
^ For example, an alloy of 30 atomic per cent Zn in Cu has an electronatom ratio
of 1.3. (3 X 2) + (7 X Ij = 13 valence electrons per 3 + 7 = 10 atoms.
2 A. L. Norbury, Trans. Faraday Soc, vol. 19, pp. 506600, 1924; R. M. Brick, D.
L. Martin, and R. P. Angier, Trans. ASM, vol. 31, pp. 675698, 1943; J. H. Frye and
W. HumeRothery, Proc. Roy. Soc. (London), vol. 181, pp. 114, 1942.
)ec.
54]
Plastic Deformation of Polycrystaliinc Aggregates 129
ness increase varies directly with the difference in the size of the sohite
and solvent atoms, or with the change in lattice parameter resulting
from the solute addition. However, it is apparent that size factor alone
cannot explain solidsolution hardening. An improvement in correlation
of data' results when the relative valence of the solute and solvent are
considered in addition to the latticeparameter distortion. The impor
tance of valence is shown in Fig. 58, where the yield stress of copper
alloys of constant lattice parameter is
plotted against the electronatom
ratio.' Further results^ show that
alloys with equal grain size, lattice
parameter, and electronatom ratio
have the same initial yield stress, but
the flow curves differ at larger strains.
Systemic studies of the effect of
solidsolution alloying additions on
the flow curve in tension have been
made for iron,* copper,^ aluminum,^
and nickel.^ For the case of iron the
solidsolutionstrengthened alloy is a
power function of the alloy addition.
Figure 59 shows the increase in ten
sile strength due to alloying additions
in iron. For a given atomic per cent of solute the increase in strength
varies inversely with the limit of solubility.
The distribution of solute atoms in a solvent lattice is not usually
completely random. There is growing evidence that solute atoms group
preferentially at dislocations, stacking faults, lowangle boundaries, and
grain boundaries. However, even in a perfect lattice the atoms would
not be completely random. For a solid solution of A and B atoms, if B
atoms tend to group themselves preferentially around other B atoms, thp
situation is called clustering. However, if a given B atom is preferentially
surrounded by A atoms, the solid solution exhibits shortrange order.
The tendency for clustering or shortrange order increases with increasing
solute additions.
1 J. E. Dorn, P. Pietrokowsky, and T. E. Tietz, Trans. AIME, vol. 188, pp. 933
943, 1950.
2 W. R. Hibbard, Jr., Trans. Met. Soc. AIME, vol. 212, pp. 15, 1958.
3 N. G. Ainslie, R. W. Guard, and W. R. Hibbard, Trans. Met. Soc. AIME, vol. 215,
pp. 4248, 1959.
^ C. E. Lacy and M. Gensamer, Trans. ASM, vol. 32, pp. 88110, 1944.
5 R. S. French and W. R. Hibbard, Jr., Trans. AIME, vol. 188, pp. 5358, 1950.
^ Dorn, Pietrokowsky, and Tietz, op. cit.
' V. F. Zackay and T. H. Hazlett, Ada Met., vol. 1, pp. 624628, 1953.
1.10 1.15 1.20
E!ectrono!om ratio
Fig. 58. Effect of electronatom ratio
on the yield stress of copper solidsolu
tion alloys. {W. R. Hibbaid, Ji.,
Trans. Met. Soc. AIME, vol. 212, p. 3,
1958.)
130 Metallurgical Fundamentals
[Chap. 5
It is likely that solidsolution hardening is not simply the result of
internal stresses due to the local lattice disturbance from randomly
dispersed solute atoms. Consider a dislocation line in a perfectly random
solidsolution lattice. On the average, there will be equal numbers of
positive and negative stress fields, due to solute atoms, acting on the
dislocation line. The net stress will be nearly zero, and the dislocation
0.4 0.7
Atomic % solute
Fig. 59. Increase in tensile strength of iron due to solidsolution alloy additions vs.
atomic per cent of alloy added. (C. E. Lacy and M. Gensamer, Trans. ASM, vol.
32, p. 88, 1944.)
will move through the lattice almost as easily as through the lattice of a
pure metal.
Following the ideas of Cottrell,^ it is generally held that hardening
from solute atoms results from the interaction of solute atoms, in the
form of "atmospheres," with dislocations. Since the atoms in the
region above a positive edge dislocation are compressed and below the
slip plane are stretched, the strain energy of distortion can be reduced by
large atoms collecting in the expanded region and small atoms collecting
in the compressed region. Interstitial atoms collect in the expanded
1 A. H. Cottrell, "Dislocations and Plastic Flow in Crystals," Oxford University
Press, New York, 1953.
Sec. 54] Plastic Deformation of Polycrystalline Aggregates 131
region below the slip plane of a positive edge dislocation. Because the
local energy is lower when a dislocation is surrounded by a solute atmos
phere, a higher stress is required to make the dislocation move than would
be required if there were no interaction between the dislocation and the
solute atoms. If the stress becomes high enough, the dislocation can be
torn away from its atmosphere. When this happens, the dislocation is
free to move at a lower stress.
The bestknown case of dislocation interaction wj^^h a soluteatom
atmosphere is the existence of an upper and lower yield point in iron and
other metals. The occurrence of a yield point in iron is known to be
associated with interstitial solute atoms (see Sec. 55). The upper yield
point corresponds to the stress required to tear dislocations away from
their atmospheres of interstitial atoms.
A number of types of soluteatom interaction must be considered in
explaining solidsolution strengthening.^ Cottrell locking due to elastic
interaction between the solute atoms and the dislocations, such as is
described above for interstitial atoms, is certainly an important factor in
solidsolution strengthening. In view of the valency effects observed in
solid solutions, electrical interaction must also be considered. However,
estimates show that electrical interaction is only about onethird to one
seventh as strong as elastic interaction. Suzuki has pointed out the
existence of a third type of interaction. Thermodynamic reasoning
shows that the concentration of solute atoms at a stacking fault will be
greater than the average bulk concentration. Thus, there is a "chemical
interaction" between these regions and dislocations. While for most
alloys this chemical interaction is weaker than the interaction force due
to Cottrell locking, the force due to chemical interaction does not decrease
with increasing temperature nearly so much as in the case of Cottrell
locking. Fisher^ has pointed out that the existence of shortrange order
or clustering in an alloy will produce a strengthening effect. Slip in a
pure metal does not change the internal energy of the lattice, because the
configuration of atoms across the slip plane is the same after slip as before.
The same situation would exist for a completely random solid solution,
but in an alloy with shortrange order slip will partially destroy the
1 Theories of solidsolution strengthening are reviewed by E. R. Parker and T. H.
Hazlett, "Relation of Properties to Microstructure," pp. 5053, American Society for
Metals, Metals Park, Ohio, 1954. A fairly mathematical discussion of the interactions
between dislocations and solute atoms is given by A. H. Cottrell, "Relation of Proper
ties to Microstructure," pp. 131162, American Society for Metals, Metals Park,
Ohio, 1954.
2 H. Suzuki, Sci. Repts. Research Insts. Tohoku Univ., vol. 4A, no. 5, pp. 455463,
1952; "Dislocations and Mechanical Properties of Crystals," p. 361, John Wiley &
Sons, Inc., New York, 1957.
3 J. C. Fisher, Ada Met., vol. 2, p. 9, 1954.
132 Metallurgical Fundamentals [Chap. 5
pattern of order across the slip plane. An internal surface of increased
energy is produced at the slip plane, and this results in an increase in
the stress required to produce slip. The chemical interaction of Suzuki
would be expected to predominate over shortrange order in dilute solu
tions, where the stackingfault energy decreases rapidly with concentra
tion. In concentrated solid solutions strengthening from shortrange
order should predominate.
In a binary alloy with longrange order each of the constituent atoms
occupies special sites in the lattice. In effect, this results in a superlattice
with a larger unit cell and a new crystal structure. The interaction of
dislocations with longrange order^ results in a strengthening effect. An
ordered crystal will contain domains within which the order is perfect,
but which are out of step with the order in the neighboring domains.
Since the domain boundaries are a highenergy interface, there is an
interaction between dislocations and these antiphase boundaries. The
stress required to produce slip varies inversely with the distance between
domain boundaries. Because more domain boundaries are produced as
slip continues, the rate of strain hardening is higher in the ordered condi
tion than in the disordered state. Ordered alloys with a fine domain size
(approximately 50 A) are stronger than the disordered state. Ordered
alloys with a large domain size generally have a yield stress lower than
that of the disordered state. This arises from the fact that the disloca
tions in a wellordered alloy are grouped into pairs, each pair having a
Burgers vector twice as large as that for the disordered lattice.
55. Yieldpoint Phenomenon
Many metals, particularly lowcarbon steel, show a localized, hetero
geneous type of transition from elastic to plastic deformation which
produces a yield point in the stressstrain curve. Rather than having a
flow curve with a gradual transition from elastic to plastic behavior, such
as was shown in Fig. 31, metals with a yield point have a flow curve or,
what is equivalent, a loadelongation diagram similar to Fig. 510. The
load increases steadily with elastic strain, drops suddenly, fluctuates
about some approximately constant value of load, and then rises with
further strain. The load at which the sudden drop occurs is called the
upper yield point. The constant load is called the lower yield point, and
the elongation which occurs at constant load is called the yieldpoint
elongation. The deformation occurring throughout the yieldpoint elon
1 N. Brown and M. Herman, Trans. AIME, vol. 206, pp. 13531354, 1954; A. H.
Cottrell, "Relation of Properties to Microstriicture," pp. 131162, American Society
for Metals, Metals Park, Ohio, 1954; N. Brown, Phil. Mag., vol. 4, pp. 693704, 1959;
P. A. Flinn, Trans. AIME, vol. 218, pp. 145154, 1960.
55]
Plastic Deformation of Polycrystalline Aggregates 133
,. Upper yield point
Elongotion
Fig. 510. Typical yieldpoint behavior.
gation is heterogeneous. At the upper jdeld pomt a discrete band of
deformed metal, often readily visible with the eye, appears at a stress
concentration such as a fillet, and coincident with the formation of the
band the load drops to the lower yield point. The band then propagates
along the length of the specimen, causing the yieldpoint elongation.
In the usual case several bands will
form at several points of stress
concentration. These bands are
generally at approximately 45° to
the tensile axis. They are usually
called Luders hands, Hartmann
lines, or stretcher strains, and this
type of deformation is sometimes
referred to as the Piobert effect.
When several Liiders bands are
formed, the flow curve during the
yieldpoint elongation will be ir
regular, each jog corresponding
to the formation of a new Liiders band. After the Liiders bands have
propagated to cover the entire length of the specimen test section, the
flow will increase with strain in the usual manner. This marks the end
of the yieldpoint elongation.
The yieldpoint phenomenon was found originally in lowcarbon steel.
A pronounced upper and lower yield point and a yieldpoint elongation of
over 10 per cent can be obtained wdth this material under proper condi
tions. More recently the yield point has come to be accepted as a general
phenomenon, since it has been observed in a number of other metals and
alloys. In addition to iron and steel, yield points have been observed in
polycrystalline molybdenum, titanium, and aluminum afloys and in
single crystals of iron, cadmium, zinc, alpha and beta brass, and alumi
num. Usually the yield point can be associated with small amounts of
interstitial or substitutional impurities. For example, it has been shown^
that almost complete removal of carbon and nitrogen from lowcarbon
steel by wethydrogen treatment will remove the yield point. However,
only about 0.001 per cent of either of these elements is required for a
reappearance of the yield point.
A number of experimental factors affect the attainment of a sharp
upper yield point. A sharp upper yield point is promoted by the use of
an elastically rigid (hard) testing machine, very careful axial alignment
of the specimen, the use of specimens free from stress concentrations, high
rate of loading, and, frequently, testing at subambient temperatures.
If, through careful avoidance of stress concentrations, the first Liiders
1 J. R. Low and M. Gensamer, Trans. AIME, vol. 158, p 207, 1944.
134 Metallurgical Fundamentals [^hap. 5
band can be made to form at the middle of the test specimen, the upper
yield point can be roughly twice the lower yield point. However, it is
more usual to obtain an upper yield point 10 to 20 per cent greater than
the lower yield point.
Cottrell's concept that the yield point is due to the interaction of solute
atoms with dislocations was introduced in the previous section. Solute
atoms diffuse to dislocations because this lowers the strain energy of the
crystal. The dislocations are then anchored in position by an atmosphere
of solute atoms. The original theory^ considered that solute atoms would
segregate only to edge dislocations, because a screw dislocation ordinarily
has no tensile component. More recently the theory has been modified
to show that there is a strong interaction between interstitial atoms and
screw dislocations when the lattice is nonsymmetrically deformed by the
solute atoms so that a tensile component of stress is developed. 
The local concentration of solute atoms near the dislocation, c, is
related to the average concentration co by the relationship
c = Co exp — T— (56)
where U is the interaction energy. For carbon and nitrogen in iron the
interaction energy has a value between 0.5 and 1.0 ev. As the tempera
ture decreases, the solute atmosphere becomes more concentrated and
below a critical temperature the atmosphere condenses into a line of
solute atoms. These atoms occupy a position of maximum interaction
energy just below the center of a positive edge dislocation running parallel
to the length of the dislocation.
The shear stress required to tear away a dislocation from its atmosphere
goes through a maximum when plotted against displacement. Therefore,
a dislocation will tend to return to its atmosphere for small displacements,
but when a certain breakaway stress has been reached, movement of
the dislocation becomes easier with increasing distance from the atmos
phere. The stress at which the dislocations break away from their
atmosphere corresponds to the upper yield point. This releases an
avalanche of dislocations into the slip plane, and these pile up at the
grain boundary. The stress concentration at the tip of the pileup
combines with the applied stress in the next grain to unlock the disloca
tions in that grain, and in this way a Liiders band propagates over the
specimen.
1 A. H. Cottrell and B. A. Bilby, Proc. Phijs. Soc. (London), vol. 62A, pp. 4962,
1949.
2 A. W. Cochardt, G. Schoek, and H. Wiedersich, Acta Met., vol. 3, pp. 533537,
1955.
Sec. 56]
Plastic DeFormation of Polycrystalline Aggregates 135
56. Strain Asing
Strain aging is a type of behavior, usually associated with the yield
point phenomenon, in which the strength of a metal is increased and the
ductility is decreased on heating at a relatively low temperature after
cold working. This behavior can best be illustrated by considering
Fig. 511, which schematically describes the effect of strain aging on the
Strain
Fig. 511. Stressstrain curves for lowcarbon steel showing strain aging. Region A,
original material strained through yield point. Region B, immediately retested after
reaching point A'. Region C, reappearance and increase in yield point after aging at
300°F.
flow curve of a lowcarbon steel. Region A of Fig. 511 shows the stress
strain curve for a lowcarbon steel strained plastically through the yield
point elongation to a strain corresponding to point X. The specimen is
then unloaded and retested without appreciable delay or any heat treat
ment (region B). Note that on reloading the yield point does not occur,
since the dislocations have been torn away from the atmosphere of carbon
and nitrogen atoms. Consider now that the specimen is strained to point
Y and unloaded. If it is reloaded after aging for several days at room
temperature or several hours at an aging temperature like 300°F, the
yield point will reappear. Moreover, the yield point will be increased
by the aging treatment from F to Z. The reappearance of the yield
point is due to the diffusion of carbon and nitrogen atoms to the dis
locations during the aging period to form new atmospheres of interstitials
anchoring the dislocations. Support for this mechanism is found in the
fact that the activation energy for the return of the yield point on aging
is in good agreement with the activation energy for the diffusion of carbon
in alpha iron.
136 Metallurgical Fundamentals [Chap. 5
Nitrogen plaj^s a more important role in the strain aging of iron than
carbon because it has a higher solubihty and diffusion coefficient and
produces less complete precipitation during slow cooling. From a
practical standpoint it is important to eliminate strain aging in deep
drawing steel because the reappearance of the yield point can lead to
difficulties with surface markings or "stretcher strains" due to the local
ized heterogeneous deformation. To control strain aging, it is usually
desirable to lower the amount of carbon and nitrogen in solution by
adding elements which will take part of the interstitials out of solution
in the form of stable carbides or nitrides. Aluminum, vanadium,
titanium, columbium, and boron have been added for this purpose.
While a certain amount of control over strain aging can be achieved,
there is no commercial lowcarbon steel which is completely nonstrain
aging. The usual industrial solution to this problem is to deform the
metal to point X by roller leveling or a skinpass rolling operation and use
it immediately before it can age.
Just as the existence of a yield point has become recognized as a general
metallurgical phenomenon, so the existence of strain aging has come to
be recognized in metals other than lowcarbon steel. In addition to the
return of the yield point and an increase in the yield point after aging,
it has been suggested^ that a serrated flow curve and a minimum in the
variation of strainrate sensitivity with temperature are characteristics
of strain aging. The strainrate sensitivity is the change in stress required
to produce a certain change in strain rate at constant temperature (see
Chap. 9). The occurrence of serrations in the stressstrain curve is
known as discontinuous, or repeated, yielding. It is also called the
PortevinLe Chdtelier effect. This phenomenon is due to successive
yielding and aging while the specimen is being tested. This results from
the fact that in the range of temperature in which it occurs the time
required for the diffusion of solute atoms to dislocations is much less
than the time required for an ordinary tension test. Discontinuous
yielding is observed in aluminum3 per cent magnesium alloys, duralumin,
alpha brass, and plaincarbon steel.
For plaincarbon steel discontinuous yielding occurs in the temperature
region of 450 to 700°F. This temperature region is known as the blue
brittle region because steel heated in this temperature region shows a
decreased tensile ductility and decreased notchedimpact resistance.
This temperature range is also the region in which steels show a minimum
in strainrate sensitivity and a maximum in the rate of strain aging. All
these facts point to the realization that blue brittleness is not a separate
phenomenon but is just an accelerated strain aging.
The phenomenon of strain aging should be distinguished from a process
» J. D. Lubahn, Trans. ASM, vol. 44, pp. 643666, 1952.
Sec. 57] Plastic Deformation of Polycrystalline Aggregates 137
known as quench aging, which occurs in lowcarbon steels. Quench
aging is a type of true precipitation hardening that occurs on quenching
from the temperature of maximum solubility of carbon and nitrogen in
ferrite. Subsequent aging at room temperature, or somewhat above,
produces an increase in hardness and yield stress, as in the age hardening
of aluminum alloys. Plastic deformation is not necessary to produce
quench aging.
57. Strengthening from Secondphase Particles
Only a relatively small number of alloy systems permit extensive solid
solubility between two or more elements, and only a relatively small
hardening effect can be produced in most alloy systems by solidsolution
additions. Therefore, most commercial alloys contain a heterogeneous
microstructure consisting of two or more metallurgical phases. A num
ber of different conditions may be encountered.^ The two phases may
be ductile and present in the microstructure in relatively massive form,
as in alphabeta brass. On the other hand, the structure may consist
of a hard, brittle phase in a ductile matrix, as in spheroidized steel or
WC particles in a cobalt matrix in a cemented carbide cutting tool.
The strengthening produced by secondphase particles is usually addi
tive to the solidsolution strengthening produced in the matrix. For
twophase alloys produced by equilibrium methods, the existence of a
second phase ensures maximum solidsolution hardening because its
presence resulted from supersaturation of the continuous phase. More
over, the presence of secondphase particles in the continuous matrix
phase results in localized internal stresses which modify the plastic
properties of the continuous phase. Many factors must be considered
for a complete understanding of strengthening from secondphase par
ticles. These factors include the size, shape, number, and distribution
of the secondphase particles, the strength, ductility, and strainhardening
behavior of the matrix and second phase, the crystallographic fit between
the phases, and the interfacial energy and interfacial bonding between
the phases. It is almost impossible to vary these factors independently
in experiments, and it is very difficult to measure many of these quantities
with any degree of precision. Therefore, our existing knowledge of the
effect of second phases on mechanical properties is mainly empirical and
incomplete.
In a multiphase alloy, each phase contributes certain things to the
overall properties of the aggregate. If the contributions from each
' A review of the effect of secondphase particles on mechanical properties has been
given by J. E. Dorn and C. D. Starr, "Relation of Properties to Microstructure,"
pp. 7194, American Society for Metals, Metals Park, Ohio, 1954.
138 Metallurgical Fundamentals
[Chap. 5
phase are independent, then the properties of the multiphase alloy will
be a weighted average of the properties of the individual phases. For
example, the density of a twophase alloy will be equal to the sum of the
volimie fraction of each phase times its density. However, for the
structuresensitive mechanical properties the properties of the aggregate
are generally influenced by interaction between the two phases. Two
simple hypotheses may be used to calculate the properties of a twophase
alloy from the properties of the individual phases. It it is assumed that
Fig. 512. Estimate of flow stress of twophase alloy, (a) Equal strain; (b) equal
stress. (From J. E. Dorn and C. D. Starr, "Relation of Properties to Microstrudure,"
pp. 7778, American Society for Metals, Metals Park, Ohio, 1954.)
the strain in each phase is equal, the average stress in the alloy for a
given strain will increase linearly with the volume fraction of the strong
phase.
Oavg = /lOl + /2Cr2 (57)
The volume fraction of phase 1 is/i, and/i + /a = 1. Figure 512a shows
the calculation of the flow curve for an alloy with 0.5 volume fraction of
phase 2 on the basis of the equalstrain hypothesis. An alternative
hypothesis is to assume that the two phases are subjected to equal stresses.
The average strain in the alloy at a given stress is then given by
= /lei + fo^i
(58)
Figure 5126 shows the flow curve for a 0.5volumefraction alloy on the
basis of the equalstress hypothesis. Both these hypotheses are simple
approximations, and the strengths of alloys containing two ductile phases
usually lie somewhere between the values predicted by the two models.
The deformation of an alloy consisting of two ductile phases depends
upon the total deformation and the volume fractions of the phases.
Slip will occur first in the weaker phase, and if very little of the stronger
Sec. 57] Plastic Deformation of Polycrystalline Aggregates 139
phase is present, most of the deformation will continue in the softer phase.
At large deformations flow of the matrix will occur around the particles
of the harder phase. If the volume fraction of the harder phase is less
than about 0.3, the soft phase deforms more than the hard phase for
reductions of up to 60 per cent. At greater reductions the two phases
deform more uniformly. When the phases are present in about equal
amounts, they deform to about the same extent.^
The mechanical properties of an alloy consisting of a ductile phase and
a hard brittle phase will depend on how the brittle phase is distributed in
the microstructure. If the brittle phase is present as a grainboundary
envelope, as in oxygenfree copperbismuth alloys or hypereutectoid
steel, the alloy is brittle. If the brittle phase is in the form of discon
tinuous particles at grain boundaries, as when oxygen is added to copper
bismuth alloys or with internally oxidized copper and nickel, the brittle
ness of the alloy is reduced somewhat. When the brittle phase is present
as a fine dispersion uniformly distributed throughout the softer matrix,
a condition of optimum strength and ductility is obtained. This is the
situation in heattreated steel with a tempered martensitic structure.
The strengthening produced by a finely dispersed insoluble second
phase in a metallic matrix is known as dispersion hardening. A very
similar strengthening phenomenon, precipitation hardening, or age harden
ing, is produced by solution treating and quenching an alloy in which a
second phase is in solid solution at the elevated temperature but precipi
tates upon quenching and aging at a lower temperature. The age
hardening aluminum alloys and copperberyllium alloys are common
examples. For precipitation hardening to occur, the second phase must
be soluble at an elevated temperature but must exhibit decreasing
solubility with decreasing temperature. By contrast, the second phase
in dispersionhardening systems has very little solubility in the matrix,
even at elevated temperatures. Usually there is atomic matching, or
coherency, between the lattices of the precipitate and the matrix, while in
dispersionhardened systems there generally is no coherency between the
secondphase particles and the matrix. The requirement of a decreasing
solubility with temperature places a limitation on the number of useful
precipitationhardening alloy systems. On the other hand, it is at least
theoretically possible to produce an almost infinite number of dispersion
hardened systems by mixing finely divided metallic powders and second
phase particles (oxides, carbides, nitrides, borides, etc.) and consolidating
them by powder metallurgy techniques. Advantage has been taken of
this method to produce dispersionhardened systems which are thermally
stable at very high temperatures. Because of the finely dispersed second
phase particles these alloys are much more resistant to recrystallization
' L. M. Clarebrough, Australian J. Sci. Repts., vol. 3, pp. 7290, 1950.
140 Metallurgical Fundamentals [Chap. 5
and grain growth than singlephase alloys. Because there is very little
solubility of the secondphase constituent in the matrix, the particles
resist growth or overaging to a much greater extent than the second
phase particles in a precipitationhardening system.
The formation of a coherent precipitate in a precipitationhardening
system, such as AlCu, occurs in a number of steps. After quenching
from solid solution the alloy contains regions of solute segregation, or
clustering. Guiner and Preston first detected this local clustering with
special Xray techniques, and therefore this structure is known as a GP
zone. The clustering may produce local strain, so that the hardness of
GP[1] is higher than for the solid
solution. With additional aging
Loss of coherency the hardness is increased further
by the ordering of larger clumps of
Equilibrium copper atoms on the {100} planes
precipitate Qf ^j^g matrix. This structure is
known as GP[2], or Q" . Next,
„ . . definite precipitate platelets of
Aging time ^
(Particle size ^) CUAI2, or 0, which are coherent
r. r 4o Tr • 4^ 4^ • ij 4^ it, with the matrix, form on the {100}
rig. 51 3. Variation 01 yield stress with ' ' '
aging time (schematic). planes of the matrix. The co
herent precipitate produces an in
creased strain field in the matrix and a further increase in hardness.
With still further aging the equilibrium phase CUAI2, or 0, is formed from
the transition lattice Q' . These particles are no longer coherent with the
matrix, and therefore the hardness is low^er than at the stage when
coherent Q' was present. For most precipitationhardening alloys the
resolution with the light microscope of the first precipitate occurs after
the particles are no longer coherent with the matrix. Continued aging
beyond this stage produces particle growth and further decrease in hard
ness. Figure 513 illustrates the way in which strength varies with aging
time or particle size. The sequence of events in the AlCu system is
particularly complicated. Although other precipitationhardening sys
tems may not have so many stages, it is quite common for a coherent
precipitate to form and then lose coherency when the particle grows to a
critical size.
Metallographic observations of deformation mechanisms in precipita
tionhardening systems require very careful techniques.^ In the as
quenched condition slip bands are broad and widely spaced. As aging
continues, the slip bands become finer and more closely spaced. As
GP[2] zones form and the alloy proceeds toward peak hardness, fewer
and fewer slip bands can be observed with the electron microscope.
1 (J. Thomas and J. Nutting, J. Insl. Melals, vol. 86, pp. 714, 19571958; R. B.
Nicholson, G. Thomas, and J. Nutting, Acta Met., vol. 8, pp. 172176, 1960.
Sec. 57] Plastic Deformation of Polycrystalline Aggregates 141
When the alloy begins to overage and coherency breaks down, the slip
lines once more can be observed. Electronmicroscope studies have
shown that dislocation motion is impeded by fully and partially coherent
precipitates, but eventually the dislocations shear Ihrouj^h the particles.
For a noncoherent precipitate the slip lines do not cut through the
particles. Instead, the dislocation lines bend to avoid the particles,
probably by a process of cross slip.^
The degree of strengthening resulting from secondphase dispersions
depends upon the distribution of particles in the soft matrix. In addi
tion to shape, the secondphase dispersion can be described by specifying
the volume fraction, average particle diameter, and mean interparticle
spacing. These factors are all interrelated so that one factor cannot be
changed without affecting the others (see Prob. 55). For example, for
a given volume fraction of second phase, reducing the particle size
decreases the average distance between particles. For a given size
particle, the distance between particles decreases with an increase in
the volume fraction of second phase. Quantitative relationships between
strength and the geometrical factors have not been determined to any
extent for real alloys.
However, the qualitative aspects of dispersion hardening can be con
sidered, the common situation of carbide particles in ferrite being used as
an example. In general, the hardness and strength increase with carbon
content or volume fraction of the carbide phase. Further, for a given
carbon content, the strength will be higher for a fine carbide spacing
than with a coarse interparticle spacing. Particle shape has a less impor
tant effect on tensile properties, although for a given volume fraction of
carbides lamellar carbides will be stronger than spheroidized carbides.
Particle shape is of greater importance in notched impact, where a
spheroidized structure will be tougher than a lamellar structure.
Detailed quantitative metallography on steels heattreated to provide
different interparticle spacings has shown the relationship between
strength and structure. Gensamer and coworkers found that the flow
stress, at a true strain of 0.2, was inversely proportional to the logarithm
of the mean interparticle spacing (mean free ferrite path) for pearlite
and spheroidite structures (Fig. 514). Confirmation of this relation
ship has been found for tempered martensitic structures' and overaged
AlCu alloys.'' Figure 515 illustrates the marked strengthening produced
by CuAl2 particles in an AlCu alloy. The figure shows the variation of
1 P. B. Hirsch, /. Inst. Metals, vol. 86, pp. 1314, 195758.
2 M. Gensamer, E. B. Pearsall, W. S. Pellini, and J. R. Low, Jr., Trans. ASM,
vol. 30, pp. 9831020, 1942.
3 A. M. Turkalo and J. R. Low, Jr., Trans. Met. Soc. AIMS, vol. 212, pp. 750758,
1958.
^C. D. Starr, R. B. Shaw, and J. E. Dorn, Trans. ASM, vol. 46, pp. 10751088,
1954.
142 Metallurgical Fundamentals
[Chap. 5
200
\
•
\
• •
%
(
•
160
^
yo
^.
oN
o
120
o
o
\
%
o
\^A
A
A
80
o
° O^
\,
\
\,
40
<niH np
nrlitp
\
° Spheroidife
^ Hypoeutecfoid pearl ne
1 1 1 1
3.0
3.4 3.8 42 4.6 5.0 5.4
log of mean ferrite path, angstroms
5.8
Fig. 514. Flow stress vs. logarithm of mean free ferrite path in steels with pearlitic
and spheroidal distribution of carbides. (M. Gensamer, E. B. Pearsall, W. S. Pelhni,
and J. R. Low, Trans. ASM, vol. 30, p. 1003, 1942.)
300 400 500
Temperature, °K
700
Fig. 515. Variation of flow stress with temperature for AICu alloy containing 5
volume per cent fine and coarse secondphase particles. (C D. Starr, R. B. Shaw,
and J. E. Darn, Trans. ASM, vol. 46, p. 1085, 1954.)
>ec.
57]
Plastic Deformation of Polycrystalline Aggresates 143
flow stress witk temperature for an alloy containing 5 volume per cent
of CuAU in three conditions. The top curve i»s for a fine CuAU disper
sion, the middle curve for a coarse dispersion, and the bottom curve is for
an alloy without dispersed particles which contains the same amount of
copper in solid solution as the top two alloys.
Table 52 gives the variation of proportional limit and tensile strength
Table 52
Variation of Tensile Properties with Volume Fraction
OF Second Phase for CoWC Alloys!
Volume
Mean
Proportional
Tensile
fraction
interparticle
limit,
strength,
of WC
path, n
psi X 103
psi X 103
0.00
103
0.10
16.8
9
102
0.35
3.4
21
165
0.50
1.7
40
173
0.63
1.0
74
193
0.78
0.4
85
124
0.90
0.2
95
t C. Nishimatsu and J. Gurland, Trans. ASM, vol. 52, pp. 469484, 1960.
with composition and mean interparticle spacing for a series of CoWC
alloys. These alloys were prepared by powder metallurgy and consisted
of uniform dispersions of 2fi WC particles in a cobalt matrix. The rapid
increase in proportional limit with increasing volume fraction of second
phase shows the effect of decreasing interparticle spacing in raising the
flow stress of the ductile matrix. Tensile strength is much less sensitive.
However, when the microstructure is nearly all tungsten carbide, the
material fails in a brittle manner by fracture through the carbides. Frac
ture initiates in the brittle carbide phase but does not propagate readily
through the surrounding cobalt envelope. However, with a high volume
fraction of carbide many WC particles are touching, and the brittle frac
ture can propagate readily from carbide to carbide. This is reflected in
a decrease in tensile strength.
Dislocation models of dispersion hardening and precipitation harden
ing consider that secondphase particles act as obstacles to the movement
of dislocations. In making the first analysis of this problem, Mott and
Nabarro^ considered that a dislocation line would take on a smoothly
curved form when moving through the lattice instead of moving as a
straight line. Since different sections of the dislocation line can move
partly independently of each other, the random stress fields in the matrix
which intc/act with the dislocation line do not cancel out. Because a
1 N. F. IMott and F. R,. N. Nabarro, Proc. Phys. Soc. {London), vol. 52, p. 86, 1940.
144
Metallurgical Fundamentals
[Chap. 5
dislocation possesses line tension, which tends to keep it at its shortest
length, any bending or increase in length of a dislocation line requires the
expenditure of extra energy. The smallest radius of curvature to which a
dislocation line can be bent under the influence of an internal stress field
Ti is given by
Gb
R =
2Ti
(59)
Orowan' suggested that the yield stress of an alloy containing a disper
sion of fine particles is determined by the shear stress required to force a
dislocation line between two particles separated by a distance A. In
'^
Ci>< ®
Ci> c ®
(1) (2) (3) (4) (5)
Fig. 516. Schematic drawing of stages in passage of a dislocation between widely
separated obstacles — based on Orowan's mechanism of dispersion hardening.
Fig. 516, stage 1 shows a straight dislocation line approaching two
particles separated by a distance A. At stage 2 the line is beginning to
bend, and at stage 3 it has reached the critical stage. Since A equals
twice the critical radius of curvature, from Eq. (59) the stress needed to
force the dislocation line between the obstacles is given by
Gb
A
(510)
At stage 4 the dislocation has passed between the obstacles, leaving them
encircled by small loops of dislocations. Every dislocation gliding over
the slip plane adds one loop around the obstacle. These dislocation
loops exert a back stress which must be overcome by dislocations moving
on the slip plane. This requires an increase in shear stress to continue
deformation. Thus, the presence of dispersed particles leads to increased
strain hardening during the period when loops are building up around the
particles. This continues until the shear stress developed by the loops is
high enough to shear the particles or the surrounding matrix. According
to the theory developed by Fisher, Hart, and Pry,^ the increase in shear
^ E. Orowan, discussion in "Symposium on Internal Stresses," p. 451, Institute of
Metals, London, 1947.
2 J. C. Fisher, E. W. Hart, and R. H. Pry, Ada Met., vol. 1, p. 33G, 1953.
Sec. 58] Plastic DeFormation of Polycrystalline Aggregates 145
stress due to fine particles, ta, is related to the volume fraction of the sec
ond phase, /, and the shear strength of a dislocationfree matrix, Tc, by
the relationship
TH = 3rJ" (511)
where n is between 1 and 1 .5.
Orowan's relationship between strength and particle spacing has been
experimentally confirmed for most systems containing overaged or non
coherent particles. The Fisher, Hart, and Pry equation for the contribu
tion to strain hardening from dispersed particles also appears to be
approximately verified. According to Eq. (510) the shear strength of a
dispersionhardened alloy will have a maximum value when A has a value
that makes it e(}ually likely that dislocations will pass between the
particles or cut through them. As the distance between particles is
increased, the critical radius of curvature is increased and the stress
required to bend the dislocation line is decreased. When the distance
between particles is decreased, the dislocation line becomes more rigid.
It is difficult for the dislocation line to bend sharply enough to pass
between the particles, and so it shears through them instead. There are
indications that in the region of small particle spacing the yield stress is a
direct function of the radius of the particles.
58. Hardening Due to Point Defects
Vacancies and interstitials are produced by the bombardment of a
metal with highenergy nuclear particles. The bombardment of the
lattice with fast neutrons having energies up to 2 million ev knocks atoms
into interstitial positions in the lattice, and vacancies are left behind.
Neutron irradiation increases the hardness and yield strength of most
metals. In copper single crystals a dose of 10'^ neutrons per scjuare
centimenter increases the yield strength by a factor of 10 and changes
the deformation characteristics so that they are similar to alpha brass.'
In metals which show a ductiletobrittle transition, such as steel, pro
longed neutron irradiation can appreciably raise the transition tempera
ture. The structural changes producing radiation hardening and radia
tion damage are difficult to study in detail because at least two point
defects are acting simultaneously. Interstitials are even more mobile
than vacancies, so that quite low temperatures are required to prevent
them from interacting with other lattice defects.
A situation in which the only point defects are vacancios can be
produced by rapidly quenching a pure metal (so that there can be no
'A. H. Cottrell, "Vacancies and Other Point Defects in Metals and Alloys,"
pp. 139, Institute of Metals, London, 1958.
146 Metallurgical Fundamentals [Chap. 5
precipitation of a second phase) from a temperature near its melting
point. At room temperature or below the metal contains a super
saturated solution of most of the vacancies that existed in equilibrium at
the higher temperature. Vacancy concentrations of up to about 10~^
can be achieved by quenching. Soft metals, such as aluminum, copper,
and zinc, can be hardened by introducing a randomly distributed popula
tion of vacancies in this way. Quench hardening results in an increase
in yield stress and a decrease in the rate of strain hardening, just as is
produced by radiation hardening. Therefore, a dispersion of point
defects can produce hardening, by analogy with the hardening produced
by a dispersion of secondphase particles. The mechanism by which this
occurs is not yet established. There is some evidence that at this stage
the single vacancies have migrated into clusters. A greater quench
hardening results if an aging treatment is interposed between the quench
and the measurement of the tensile properties. It is likely that the aging
permits the vacancies to migrate to dislocations, where they interact and
impede the movement of dislocations (see Sec. 612). Much remains to
be learned about the interaction of point defects with each other and with
line defects and how these interactions affect the mechanical properties.
Plastic deformation produces point defects, chiefly vacancies. These
point defects are created by the intersection of dislocations, and therefore,
a discussion of this topic will be deferred until Chap. 6. Vacancy forma
tion appears to be particularly important in the fatigue of metals, and it
will be considered from this standopint in Chap. 12. At elevated tem
peratures vacancies become very important in controlling diffusion and in
making possible dislocation climb. Thus, vacancies are important in the
creep of metals, and they will be considered in greater detail in Chap. 13.
59. Strain Hardening and Cold Work
In Chap. 4 strain hardening was attributed to the interaction of dis
locations with other dislocations and with other barriers to their motion
through the lattice. So long as slip takes place on only a single set of
parallel planes, as with single crystals of hep metals, only a small amount
of strain hardening occurs. However, even with single crystals extensive
easy glide is not a general phenomenon, and with polycrystalline speci
mens it is not observed. Because of the mutual interference of adjacent
grains in a polycrystalline specimen multiple slip occurs readily, and there
is appreciable strain hardening. Plastic deformation which is carried out
in a temperature region and over a time interval such that the strain
hardening is not relieved is called cold work.
Plastic deformation produces an increase in the number of dislocations,
>ec.
59]
Plastic Deformation of Polycrystallinc Asgregates 147
which by virtue of their interaction results in a higher state of internal
stress. An annealed metal contains about 10® to 10^ dislocations per
square centimeter, while a severely plastically deformed metal contains
about 10^^ dislocations per square centimeter. Strain hardening or cold
work can be readily detected by Xray diffraction, although detailed
analysis of the Xray patterns in terms of the structure of the coldworked
state is not usually possible. In Laue patterns cold work produces a
blurring, or asterism, of the spots. For DebyeScherrer patterns the lines
are broadened by cold work. Xray
Regions of relatively
perfect lottice
Groin boundary
Distorted
regions of
high dislocation
density
Fig. 517. Model of the structure of cold
worked metal (schematic).
line broadening can be due to both
a decrease in size of the diffraction
unit, as would occur if the grains
were fragmented by cold work, and
an increase in lattice strain due
to dislocation interaction. Tech
niques for analyzing the entire
peak profile of Xray lines and
separating out the contribution
due to lattice strain and particle
size have been developed.^ It is
likely that improvements in this
method and more widespread application of the technique will result in
better understanding of the structure of coldworked metal.
A fairly reliable model of the structure of coldworked metal has devel
oped from microbeam Xray studies^ and from electron microscopy of
thin films. Figure 517 is a schematic drawing of the coldworked struc
ture that occurs within a single grain. It is a celllike structure consisting
of relatively perfect regions of the lattice which are connected with each
other by boundaries of dislocation networks. According to this model the
dislocation density varies drastically from a high value in the distorted
boundaries to a low value in the relatively perfect regions. The study of
the dislocation structure of coldworked metal with thinfilm electron
microscopy is a very active area of research which should provide valuable
information about how these networks vary with composition, deforma
tion, and temperature.
Most of the energy expended in deforming a metal by cold working is
converted into heat. However, roughly about 10 per cent of the expended
IB. E. Warren and B. L. Averbach, J. Appl. Phijs., vol. 21, p. 595, 1950; B. E.
Warren and B. L. Averbach, "Modern Research Techniques in Physical Metallvirgy,"
American Society for Metals, Metals Park, Ohio, 1953; B. E. Warren, "Progress in
Metal Physics," vol. 8, pp. 147202, Pergamon Press, Ltd., London, 1959.
2 P. Gay, P. B. Hirsch, and A. Kelly, Acta Cryst., vol. 7, p. 41, 1954.
148 Metallurgical Fundamentals [Chap. 5
energy is stored in the lattice as an increase in internal energy. Reported
values of stored energy^ range from about 0.01 to 1.0 cal/g of metal. The
magnitude of the stored energy increases with the melting point of the
metal and with solute additions. For a given metal the amount of stored
energy depends on the type of deformation process, e.g., wire drawing vs.
tension. The stored energy increases with strain up to a limiting value
corresponding to saturation. It increases with, decreasing temperature of
deformation. Very careful calorimeter measurements are required to
measure the small amounts of energy stored by cold working.
The major part of the stored energy is due to the generation and inter
action of dislocations during cold working. Vacancies account for part
of the stored energy for metals deformed at very low temperature. How
ever, vacancies are so much more mobile than dislocations that they
readily escape from most metals deformed at room temperature. Stack
ing faults and twin faults are probably responsible for a small fraction of
the stored energy. A reduction in shortrange order during the deforma
tion of solid solutions may also contribute to stored energy. Elastic
strain energy accounts for only a minor part of the measured stored
energy.
Strain hardening or cold working is an important industrial process that
is used to harden metals or alloys that do not respond to heat treatment.
The rate of strain hardening can be
gaged from the slope of the flow
curve. In mathematical terms,
the rate of strain hardening can be
expressed by the strainhardening
coefficient n in Eq. (31). Gener
ally, the rate of strain hardening
is lower for hep metals than for
cubic metals. Increasing tempera
ture also lowers the rate of strain
hardening. For alloys strength
ened by solidsolution additions
10 20 30 40 50 60 70 ,, . r . • i i •
Reduction by cold work, % the rate ot straui hardenmg may
Fig. 518. Variation of tensile properties ^e either increased or decreased
with amount of cold work. compared with the behavior for the
pure metal. However, the final
strength of a coldworked solidsolution alloy is almost always greater
than that of the pure metal coldworked to the same extent.
Figure 518 shows the typical variation of strength and ductility
' For a comprehensive review of the stored energy of cold work see A. L. Titchener
and M. B. Bever, "Progress in Metal Physics," vol. 7, pp. 247338, Pergamon Press,
Ltd., London, 1958.
510]
Plastic Deformation of Polycrystallinc Aggregates 149
parameters with increasing amount of cold work. Since in most cold
working processes one or two dimensions of the metal are reduced at the
expense of an increase in the other dimensions, cold work produces
elongation of the grains in the principal direction of working. Severe
deformation produces a reorientation of the grains into a preferred
orientation (Sec. 511). In addition to the changes in tensile properties
shown in Fig. 518, cold working produces changes in other physical prop
erties. There is usually a small decrease in density of the order of a few
tenths of a per cent, an appreciable decrease in electrical conductivity due
to an increased number of scattering centers, and a small increase in the
thermal coefficient of expansion. Because of the increased internal energy
of the coldworked state chemical reactivity is increased. This leads to a
general decrease in corrosion resistance and in certain alloys introduces
the possibility of stresscorrosion cracking.
510. Bauschinger Effect
In an earlier discussion of the strain hardening of single crystals it was
shown that generally a lower stress is required to reverse the direction of
slip on a certain slip plane than to
continue slip in the original direction.
The directionality of strain hardening
is called the Bauschinger effect. Fig
ure 519 is an example of the type
of stressstrain curve that is obtained
when the Bauschinger effect is
considered.
The initial yield stress of the ma
terial in tension is A. If the same
ductile material were tested in com
pression, the yield strength would be
approximately the same, point B on
the dashed curve. Now, consider
that a new specimen is loaded in
tension past the tensile yield stress to
C along the path OAC. If the speci
men is then unloaded, it will follow the
path CD, small elastichysteresis effects being neglected. If now a
compressive stress is applied, plastic flow will begin at the stress corre
sponding to point E, which is appreciably lower than the original compres
sive yield stress of the material. While the yield stress in tension was
increased by strain hardening from A to C, the yield stress in compression
was decreased. This is the Bauschinger effect. The phenomenon is
Fig. 51 9. Bauschinger effect and hys
teresis loop.
150 Metallurgical Fundamentals [Chap. 5
reversible, for had the specimen originally been stressed plastically in
compression, the yield stress in tension would have been decreased.
One way of describing the amount of Bauschinger effect is by the Bau
schinger strain 13 (Fig. 519). This is the difference in strain between the
tension and compression curves at a given stress.
If the loading cycle in Fig. 519 is completed by loading further in
compression to point F, then unloading, and reloading in tension, a
mechanicalhysteresis loop is obtained. The area under the loop will
depend upon the initial overstrain beyond the yield stress and the number
of times the cycle is repeated. If the cycle is repeated many times, failure
by fatigue is likely to occur.
Orowan^ has pointed out that, if the Bauschinger effect is due solely to
the effect of back stresses, the flow curve after reversal of strain ought
always to be softer than the flow curve for the original direction of strain.
However, not all metals show a permanent softening after strain reversal,
and those which do show only a small effect. Therefore, Orowan con
siders that the Bauschinger effect can be explained by the same mecha
nism which he proposed for dispersion hardening (Sec. 57). Obstacles
to dislocation motion are considered to be other dislocations, inclusions,
precipitate particles, etc. The stress required to move a dislocation
through these obstacles is given approximately by Eq. (510). For a
given shear stress a dislocation line will move over the slip plane until it
meets a row of obstacles that are strong enough to resist shearing and
close enough to resist the dislocation loop from squeezing between them.
Now, when the load is removed, the dislocation line will not move appreci
ably unless there are very high back stresses. However, when the direc
tion of loading is reversed, the dislocation line can move an appreciable
distance at a low shear stress because the obstacles to the rear of the dis
location are not likely to be so strong and closely packed as those imme
diately in front of the dislocation. As the dislocation line moves, it
encounters, on the average, stronger and closer obstacles, so that the
shear stress continuously increases with strain. This is in agreement
with the type of flow curve usually observed for the Bauschinger effect.
511. Preferred Orientation
A metal which has undergone a severe amount of deformation, as in
rolling or wire drawing, will develop a preferred orientation, or texture, in
which certain crystallographic planes tend to orient themselves in a pre
ferred manner with respect to the direction of maximum strain. The
tendency for the slip planes in a single crystal to rotate parallel to the axis
1 E. Orowan, Causes and Effects of Internal Stresses, in "Internal Stresses and
Fatigue in Metals," Elsevier Publishing Company, New York, 1959.
Sec. 511] Plastic Deformation of Polycrystalline Aggregates 151
of principal strain was considered in the previous chapter. The same
situation exists in a polycrystalline aggregate, but the complex inter
actions between the multiple slip systems makes analysis of the poly
crystalline situation much more difficult. Since the individual grains in
a polycrystalline aggregate cannot rotate freely, lattice bending and
fragmentation will occur.
Preferred orientations are determined by Xray methods. The Xray
pattern of a finegrained randomly oriented metal will show rings corre
sponding to different planes where the angles satisfy the condition for
Bragg reflections. If the grains are randomly oriented, the intensity of
the rings will be uniform for all angles, but if a preferred orientation
exists, the rings will be broken up into short arcs, or spots. The dense
areas of the Xray photograph indicate the orientation of the poles of the
planes corresponding to the diffraction ring in question. The orientation
of the grains of a particular crystallographic orientation with respect to
the principal directions of working is best shown by means of a pole
figure. For a description of the methods of determining pole figures and
a compilation of pole figures describing the deformation textures in many
metals, the reader is referred to Barrett.^ The current use of Geiger
counter Xray diffractometer techniques' has made it possible to deter
mine pole figures with greater accuracy and less labor than with older
film methods.
A preferred orientation can be detected with X rays after about a 20 to
30 per cent reduction in crosssectional area by cold working. At this
stage of reduction there is appreciable scatter in the orientation of indi
vidual crystals about the ideal orientation. The scatter decreases with
increasing reduction, until at about 80 to 90 per cent reduction the pre
ferred orientation is essentially complete. The type of preferred orienta
tion, or deformation texture, which is developed depends primarily on
the number and type of slip systems available and on the principal
strains. Other factors which may be important are the temperature of
deformation and the type of texture present prior to deformation.
The simplest deformation texture is produced by the drawing or rolling
of a wire or rod. This is often referred to as a fiber texture because of its
similarity to the arrangement in naturally fibrous materials. It is
important to note that a distinction should be made between the crystal
lographic fibering produced by crystallographic reorientation of the grains
during deformation and mechanical fibering, which is brought about by
the alignment of inclusions, cavities, and secondphase constituents in the
1 C. S. Barrett, "Structure of Metals," 2d ed., chap. 9, McGrawHill Book Com
pany, Inc., New York, 1952.
2 A. H. Geisler, "Modern Research Techniques in Physical Metallurgy," American
Society for Metals, Metals Park, Ohio, 1953.
152 Metallurgical Fundamentals [Chap. 5
main direction of mechanical working. Mechanical and crystallographic
fibering are important factors in producing directional mechanical
properties of plastically worked metal shapes such as sheet and rods.
This will be discussed further in Chap. 9.
In an ideal wire texture a definite crystallographic direction lies
parallel to the wire axis, and the texture is symmetrical around the wire or
fiber axis. Several types of deviations from the ideal texture are observed.
In facecentered cubic metals a double fiber texture is usually observed.
The grains have either (111) or (100) parallel to the wire axis and have
random orientations around the axis. ^ Bodycentered cubic metals have a
simple (1 10) wire texture. The wire texture in hep metals is not so simple.
For moderate amounts of deformation the hexagonal axis (0001) of zinc is
parallel to the fiber axis, while for severe deformation the hexagonal axis is
about 20° from the wire axis. For magnesium and its alloys (1010) is
parallel to the wire axis for deformation below 450° C, while above this
temperature (2110) is parallel to the fiber axis.
The deformation texture of a sheet produced by rolling is described by
the crystallographic planes parallel to the surface of the sheet as well as
the crystallographic directions parallel to the direction of rolling. There
is often considerable deviation from the ideal texture, so that pole figures
are useful for describing the degree of preferred orientation.^ Precision
determination of the rolling texture in fee metals has shown that the
texture may be described best by the {123} planes lying parallel to the
plane of the sheet with the (112) direction parallel to the rolling direction.^
This texture changes to the more common {110} (112) texture by the addi
tion of solidsolution alloying elements. In bcc metals the {100} planes
tend to be oriented parallel to the plane of the sheet with the (110) direc
tion within a few degrees of the rolling direction. For hep metals the
basal plane tends to be parallel with the rolling plane with (2110) aligned
in the rolling direction.
The preferred orientation resulting from deformation is strongly
dependent on the slip and twinning systems available for deformation, but
it is not generally affected by processing variables such as die angle, roll
diameter, roll speed, and reduction per pass. The direction of flow is the
most important process variable. For example, the same deformation
texture is produced whether a rod is made by rolling, drawing, or swaging.
The formation of a strong preferred orientation will result in an
^ It has been suggested that a (111) texture is favored by easy cross sUp, which
occurs most readily in metals with high stackingfault energy. See N. Brown,
Trans. AIME, vol. 221, pp. 236238, 1961.
2 A large number of pole figures for rolling textures are given by Barrett, op. cit.,
chap. 18.
3 R. E. Smallman, J. Inst. Metals, vol. 84, pp. 1018, 195556.
)ec.
512]
Plastic DcFormation of Polycrystalline Aggregates 153
anisotropy in mechanical properties. Although the individual grains of a
metal are anisotropic with respect to mechanical properties, when these
grains are combined in a random fashion into a polycrystalline aggregate
the mechanical properties of the aggregate tend to be isotropic. How
ever, the grain alignment that accounts for the preferred orientation
again introduces an anisotropy in mechanical properties. Different
mechanical properties in different directions can result in uneven response
of the material during forming and fabrication operations.
512. Annealins of Coldworked Metal
The coldworked state is a condition of higher internal energy than the
undeformed metal. Therefore, there is a tendency for strainhardened
Reco\/ery i Recrystollization  Grain growth
Temperature — *
Fig. 520. Schematic drawing indicating recovery, recrystallization, and grain growth
and the chief property changes in each region.
metal to revert to the strainfree condition. With increasing temperature
the coldworked state becomes more and more unstable. Eventually the
metal softens and reverts to a strainfree condition. The overall process
by which this occurs is known as annealing.' Annealing is very impor
tant commercially because it restores the ductility to a metal that has
been severely strainhardened. Therefore, by interposing annealing
operations after severe deformation it is possible to deform most metals to
a very great extent.
The overall process of annealing can be divided into three fairly distinct
processes, recovery, recrystallization, and grain growth. Figure 520 will
help to distinguish between these processes. Recovery is usually defined
as the restoration of the physical properties of the coldworked metal
without any observable change in microstructure. Electrical con
^ For detailed reviews of annealing, see P. A. Beck, Adv. in Phys., vol. 3, pp. 245
324, 1954; J. E. Burke and D. Turnbull, "Progress in Metal Physics," vol. 3, Inter
science Publishers, Inc., New York, 1952.
154 Metallurgical Fundamentals [Chap. 5
ductivity increases rapidly toward the annealed value during recovery,
and lattice strain, as measured with X rays, is appreciably reduced. The
properties that are most affected by recovery are those which are sensitive
to point defects. The strength properties, which are controlled by dis
locations, are not affected at recovery temperatures. An exception to
this is single crystals of hep metals which have deformed on only one set
of planes (easy glide) . For this situation it is possible to recover com
pletely the yield stress of a strainhardened crystal without producing
(a) (b) (D
Fig. 521. Changes in microstructure of coldworked 7030 brass with annealing.
(a) Coldworked 40 per cent; (6) 400°C, 15 min; (c) 575°C, 15min. 150X. {Courtesy
L. A. Monson.)
recrystallization. Recrystallization is the replacement of the cold
worked structure by a new set of strainfree grains. Recrystallization is
readily detected by metallographic methods and is evidenced by a
decrease in hardness or strength and an increase in ductility. The density
of dislocations decreases considerably on recrystallization, and all effects
of strain hardening are eliminated. The stored energy of cold work is the
driving force for both recovery and recrystallization. Polygonization
(Sec. 53) can be considered an intermediate situation between recovery
and recrystallization. If the new strainfree grains are heated at a
temperature greater than that required to cause recrystallization, there
will be a progressive increase in grain size. The driving force for grain
growth is the decrease in free energy resulting from a decreased grain
boundary area due to an increase in grain size. Figure 521 shows the
progression from a coldworked microstructure to a fine recrystallized
grain structure, and finally to a larger grain size by grain growth.
Recrystallization is the reversion by thermal activation of the cold
worked structure to its original strainfree condition. As the temperature
is increased, the dislocation networks tend to contract and the regions
of initially low dislocation density begin to grow. The fraction of the
microstructure that has recrystallized in a time t can be represented by an
Sec. 512] Plastic Deformation of Polycrystalline Aggregates 155
equation of the form
X = 1  exp iBt"') (512)
where B and n' are constants. Values of n' between 1 and 2 indicate one
dimensional recrystallization, while values between 2 and 3 denote two
dimensional recrystallization. It is convenient to consider the process of
recrystallization in terms of the rate of nucleation A^ and the rate of growth
G of new strainfree grains. The relative values of N and G determine the
recrystallized grain size. If A^ is large with respect to G, there are many
sites of nucleation and the grain size will be relatively small.
Six main variables influence recrystallization behavior. They are
(1) amount of prior deformation, (2) temperature, (3) time, (4) initial
grain size, (5) composition, and (6) amount of recovery or polygonization
prior to the start of recrystallization. Because the temperature at which
recrystallization occurs depends on the above variables, it is not a fixed
temperature in the sense of a melting temperature. For practical con
siderations a recrystallization temperature can be defined as the tem
perature at which a given alloy in a highly coldworked state completely
recrystallizes in 1 hr. The relationship of the above variables to the
recrystallization process can be summarized^ as follows.
1 . A minimum amount of deformation is needed to cause recrystalliza
tion.
2. The smaller the degree of deformation, the higher the temperature
required to cause recrystallization.
3. Increasing the annealing time decreases the recrystallization
temperature. However, temperature is far more important than time.
Doubling the annealing time is approximately equivalent to increasing
the annealing temperature 10°C.
4. The final grain size depends chiefly on the degree of deformation and
to a lesser extent on the annealing temperature. The greater the degree
of deformation and the lower the annealing temperature, the smaller the
recrystallized grain size.
5. The larger the original grain size, the greater the amount of cold
work recjuired to produce an equivalent recrystallization temperature.
6. The recrystallization temperature decreases with increasing purity
of the metal. Solidsolution alloying additions always raise the recrystal
lization temperature.
7. The amount of deformation required to produce equivalent recrystal
lization behavior increases with increased temperature of working.
8. For a given reduction in cross section, different metalworking proc
esses, such as rolling, drawing, etc., produce somewhat different effective
» R. F. Mehl, Recrystallization, in "Metals Handbook," pp. 259268, American
Society for Metals, Metals Park, Ohio, 1948.
156 Metallurgical Fundamentals [Chap. 5
deformations. Therefore, identical recrystallization behavior may not be
obtained.
Because the driving force for grain growth is appreciably lower than
the driving force for recrystallization, at a temperature at which recrystal
lization occurs readily grain growth will occur slowly. However, grain
growth is strongly temperaturedependent, and a graincoarsening region
will soon be reached in which the grains increase in size very rapidly.
Grain growth is inhibited considerably by the presence of a fine dispersion
of secondphase particles, which restricts grainboundary movement.
For the usual type of grain growth, where the grains increase in size uni
formly, theory predicts that at a given temperature the grain size Z> at a
time t is given by
/)2  D^ = Ct (513)
However, most experimental data agree best with an equation
where n varies from about 0.2 to 0.5, depending on the metal and the
temperature.
Under certain conditions, some of the grains of a finegrained recrystal
lized metal will begin to grow rapidly at the expense of the other grains
when heated at a higher temperature. This phenomenon is known as
exaggerated, or abnormal, grain growth. The driving force for exaggerated
grain growth is the decrease in surface energy, not stored energy, but
because the phenomenon shows kinetics similar to those of recrystalliza
tion it is often called secondary recrystallization.
51 3. Annealing Textures
The recrystallization of a coldworked metal may produce a preferred
orientation which is different from that existing in the deformed metal.
This is called an annealing texture, or recrystallization texture. An out
standing example is the cube texture in copper, where the { 100} plane lies
parallel to the rolling plane with a (001) direction parallel to the direction
of rolling. The existence of a recrystallization texture depends on a
preferential orientation of the nuclei of the recrystallized grains. Anneal
ingtexture formation depends on a number of processing variables, the
amount and type of deformation preceding annealing, the composition of
the alloy, the grain size, the annealing temperature and time, and the
preferred orientation produced by the deformation.
Generally the factors which favor the formation of a fine recrystallized
grain size also favor the formation of an essentially random orientation of
recrystallized grains. Moderate cold reductions and low annealing
Sec. 513] Plastic Deformation oF Polycrystalline Aggregates 157
temperatures are beneficial. A good way of minimizing a recrystalliza
tion texture is first to produce a strong preferred orientation by a heavy
initial reduction and then use a high anneaUng temperature. This is fol
lowed by enough added cold reduction to break up this orientation and
produce a fine recrystallized grain size at a low temperature.
Sometimes the formation of a strong recrj stallization texture is bene
ficial. The best example is cubeoriented siliconiron transformer sheet,
where the grains are oriented in the easy direction of magnetization. To
obtain a nearly perfect recrystallization texture, it is necessary to produce
a high degree of preferred orientation in the coldworked metal. This is
followed by long annealing at a high temperature to allow selective grain
growth to produce a strong texture.
BIBLIOGRAPHY
Barrett, C. S.: "Structure of Metals," 2d ed., chap. 15, McGrawHill Book Company,
Inc., New York, 1952.
Birchenall, C. E.: "Physical Metallurgy," McGrawHill Book Company, Inc., New
York, 1959.
Chalmers, B.: "Physical Metallurgy," John Wiley & Sons, Inc., New York, 1959.
Guy, A. G.: "Elements of Physical Metallurgy," 2d ed., AddisonWesley Publishing
Company, Reading, Mass., 1959.
"Relation of Properties to Microstructure," American Society for Metals, Metals
Park, Ohio, 1954.
Chapter 6
DISLOCATION THEORY
61. Introduction
A dislocation is the linear lattice defect that is responsible for nearly
all aspects of the plastic deformation of metals. This concept was
introduced in Chap. 4, where the geometry of edge and screw dislocations
was presented for the case of a simple cubic lattice. It was shown that
the existence of a dislocationlike defect is necessary to explain the low
values of yield stress observed in real crystals. A general picture has
been given of the interaction of dislocations with foreign atoms, precipitate
particles, and other dislocations. This has been used to give a qualitative
picture of the strain hardening of single crystals and, in Chap. 5, to help
explain solidsolution hardening, dispersedphase hardening, yieldpoint
behavior, and strain aging.
This chapter is intended to present a more complete and somewhat more
rigorous treatment of dislocation theory. The rapidly improving tech
niques for detecting dislocations in real metals are considered, and experi
mental evidence to support the theory is given wherever possible in sub
sequent portions of the chapter. The effect on dislocation behavior of
considering real fee, bcc, or hep crystal structures is considered. Inter
action of dislocations with other dislocations, vacancies, and foreign
atoms is discussed in some detail. The important problem of dislocation
multiplication by means of the FrankRead source is given particular
attention.
62. Methods of Detectins Dislocations
The concept of the dislocation was proposed independently by Taylor,
Orowan, and Polan^d^ in 1934, but the idea lay relatively undeveloped
until the end of World War II There followed a period of approximately
10 years in which the theory of dislocation behavior was developed
1 G. I. Taylor, Proc. Roy. Soc. (London), vol. USA, p. 362, 1934; E. Orowan, Z.
Physik, vol. 89, pp. 605, 614, 634, 1934; M. Polanyi, Z. Physik, vol. 89, p. 660, 1934.
158
Sec. 62] Dislocation Theory 159
extensively and applied to practically every aspect of the plastic deforma
tion of metals. Because there were no really reliable methods for detect
ing dislocations in real materials, it was necessary to build much of this
theory on the basis of indirect observations of dislocation behavior.
Fortunately, since 1955 improved techniques have made it possible to
observe dislocations as they actually exist in many materials. Today,
there is no question as to the existence of lattice defects with properties
similar to those ascribed to the dislocation. Many of the theoretical
predictions have been confirmed by experiment, while others have had to
be modified and some abandoned. Undoubtedly, better experimental
techniques, applicable to a wider variety of materials, will be developed
in the future. As more information is obtained on dislocation behavior in
real materials, there certainly will be other changes in current concepts of
dislocation theory.
The resolving power of the best electron microscope would have to be
improved by a factor of 5 to 10 in order to observe directly the distortion
of the individual lattice planes around a dislocation in a metal crystal.^
Practically all the experimental techniques for detecting dislocations
utilize the strain field around a dislocation to increase its effective size.
These experimental techniques can be roughly classified into two cate
gories, those involving chemical reactions with the dislocation, and those
utilizing the physical changes at the site of a dislocation. Chemical
methods include etchpit techniques and precipitation techniques. Meth
ods based on the physical structure at a dislocation site include trans
mission electron microscopy of thin films and Xray diffraction techniques.
The simplest chemical technique is the use of an etchant which forms
a pit at the point where a dislocation intersects the surface. Etch pits
are formed at dislocation sites because the strain field surrounding the
dislocation causes preferential chemical attack. A great deal of informa
tion about dislocation behavior in the ionic crystal LiF has been obtained
in this way by Oilman and Johnston.'^ Important information about
' It has been possible by means of an electron microscope to observe this lattice dis
tortion in an organic crystal of platinum phthalocyanine, which has a very large lattice
spacing (12 A) [J. W. Menter, Proc. Roy. Soc. (London), vol. 236A, p. 119, 1956]. An
indication of the lattice distortion at a dislocation in metals has been obtained by
making use of the magnification resulting from moire patterns produced by electron
transmission through two thin overlapping crystals with slightly different orienta
tions or lattice spacings. See G. A. Bassett, J. W. Menter, and D. W. Pashley, Proc.
Roy. Soc. (London), vol. 246A, p. 345, 1958.
^ Several excellent reviews of experimental techniques have been published. See
P. B. Hirsch, Met. Reviews, vol. 4, no. 14, pp. 101140, 1959; J. Nutting, Seeing Dis
locations, in "The Structure of Metals," Institution of Metallurgists, Interscience
Publishers, Inc., New York, 1959.
^J. J. Oilman and W. G. Johnston, "Dislocations and Mechanical Properties of
Crystals," John Wiley & Sons, Inc., New York, 1957.
160 Metallurgical Fundamentals
[Chap. 6
dislocations in metals has also been obtained with etchpit techniques.
Figure 61 shows the excellent resolution obtainable from etchpit studies
on alpha brass. ^ Pits only 500 A apart have been resolved. In the
region of heavy slip shown in this electron micrograph the dislocation
density is 10^" cm^^.
In metals, etchpit formation at dislocations appears to be dependent
on purity.^ Because of solute segregation to the dislocation, the region
»/j.... ., .
^J>..
'*•};
•> .( . :
M^ i
'*» ir.
a .'V'...*
■•*.., l'*"*Vt
^.
i' * ■
f*J 1 ,■:, '♦ "' "■*
Fig. 61 . Etch pits on slip bands in alpha brass crystals. 5,000 X. {J ■ D. Meakin and
H. G. F. Wilsdorf, Trans. AIME, vol. 218, p. 740, 1960.)
around the dislocation becomes anodic to the surrounding metal, and
consequently preferential etching occurs at the dislocation. Figure 54
shows an etchpit structure in an ironsilicon alloy which was made visible
by diffusion of carbon atoms to the dislocations. Etchpit techniques are
useful because they can be used with bulk samples. However, care must
be taken to ensure that pits are formed only at dislocation sites and that
all dislocations intersecting the surface are revealed.
A similar method of detecting dislocations is to form a visible precipitate
along the dislocation lines. Usually a small amount of impurity is added
to form the precipitate after suitable heat treatment. The procedure is
often called "decoration" of dislocations. This technique was first used
1 J. D. Meakin and H. G. F. Wilsdorf, Trans. AIME, vol. 218, pp. 737745, 1960.
2 A summary of etchpit techniques in metals is given by L. C. Lowell, F. L. Vogel,
and J. H. Wernick, Metal Prog., vol. 75, pp. 9696D, 1959.
Sec. 62] Dislocation Theory 161
by Hedges and MitchelP to decorate dislocations in AgBr with photolytic
silver. It has since been used with many other ionic crystals,^ such as
AgCl, NaCl, KCl, and CaF2. With these optically transparent crystals
this technique has the advantage that it shows the internal structure of
the dislocation lines. Figure 62 shows a hexagonal network of disloca
tions in a NaCl crystal which was made visible by decoration. Although
dislocation decoration has not been used extensively with metals, some
'^ X^''^^
Fig 62. Hexagonal network of dislocations in XaCl detected by a decoration technique.
(S. Amelinckx, in "Dislocations and Mechanical Properties of Crystals," John Wiley
& Sons, Inc., New York, 1957.)
work has been done along these lines with the AlCu precipitation
hardening system and with silicon crystals.
The most powerful method available today for the detection of dis
locations in metals is transmission electron microscopy of thin foils.*
Thin sheet, less than 1 mm thick, is thinned after deformation by electro
polishing to a thickness of about 1,000 A. At this thickness the specimen
is transparent to electrons in the electron microscope. Although the
crystal lattice cannot be resolved, individual dislocation lines can be
observed because the intensity of the diffracted electron beam is altered
by the strain field of the dislocation. By means of this technique it has
been possible to observe dislocation networks (Fig. 63), stacking faults,
dislocation pileup at grain boundaries (Fig. 51), CottrellLomer barriers,
and many other structural features of dislocation theory. Dislocation
1 J. M. Hedges and J. W. Mitchell, Phil. Mag., vol. 44, p. 223, 1953.
^ S. Amelinckx, "Dislocations and Mechanical Properties of Crystals," John Wiley
& Sons, Inc., New York, 1957.
5 P. B. Hirsch, R. W. Home, and M. J. Whelan, Phil. Mag., vol. 1, p. 677, 1956;
W. BoUmann, Phys. Rev., vol. 103, p. 1588, 1956.
162 Metallurgical Fundamentals [Chap. 6
movement has been observed by generating thermal stresses in the thin
foil with the electron beam. It is expected that much more information
will be gained with this method as techniques for preparing and deforming
thin foils are improved.
The dislocation structure of a crystal can be detected by Xray
diffraction microradiographic techniques.^ The strain field at the dis
location results in a different diffracted intensity. The method has the
Fig. 63. Dislocation network in coldworked aluminum. 32,500 X. (P. B. Hirsch,
R. W. Home, and M. J. Whelan, Phil. Mag., ser. 8, vol. 1, p. 677, 1956.)
advantage of being nondestructive and giving information on a bulk
sample. However, with the resolution at present available it is limited to
crystals of low dislocation density (approximately lO^cm^).
63. Burgers Vector and the Dislocation Loop
The Burgers vector b is the vector which defines the magnitude and
direction of slip. Therefore, it is the most characteristic feature of a dis
location. It has already been shown that for a pure edge dislocation the
Burgers vector is perpendicular to the dislocation line, while for a pure
screw dislocation the Burgers vector is parallel to the dislocation line.
Actually, dislocations in real crystals are rarely straight lines and rarely
lie in a single plane. In general, a dislocation will be partly edge and
partly screw in character. As shown by Figs. 62 and 63, dislocations will
1 A. R. Lang, /. Appl. Phys., vol. 30, pp. 17481755, 1959.
Sec. 63]
Dislocation Theory 163
ordinarily take the form of curves or loops, which in three dimensions
form an interlocking dislocation network. In considering a dislocation
loop in a slip plane any small segment of the dislocation line can be
resolved into edge and screw components. For example, in Fig. 64, the
dislocation loop is pure screw at point A and pure edge at point B, while
along most of its length it has mixed edge and screw components. Note,
however, that the Burgers vector is the same along the entire dislocation
loop. If this were not so, part of the crystal above the slipped region
would have to slip by a different
Slip plane
Burgers
vector
Fig. 64. Dislocation loop lying in a slip
plane (schematic).
amount relative to another part of
the crystal and this would mean
that another dislocation line would
run across the slipped region.
A convenient way of defining the
Burgers vector of a dislocation is by
means of the Burgers circuit. Con
sider Fig. 48, which shows the
atomic arrangement around an edge
dislocation. Starting at a lattice
point, imagine a path traced from
atom to atom, an equal distance in
each direction, always in the direc
tion of one of the vectors of the unit
cell. If the region enclosed by the path does not contain a dislocation,
the Burgers circuit will close. However, if the path encloses a disloca
tion, the Burgers circuit will not close. The closure failure of the Burgers
circuit is the Burgers vector b. The closure failure of a Burgers circuit
around several dislocations is equal to the sum of their separate Burgers
vectors.
Because a dislocation represents the boundary between the slipped and
unslipped region of a crystal, topographic considerations demand that it
either must be a closed loop or else must end at the free surface of the
crystal. In general, a dislocation line cannot end inside of a crystal.
The exception is at a node, where three or four dislocation lines meet. A
node can be considered as two dislocations with Burgers vectors bi and b2
combining to produce a resultant dislocation bs. The vector bs is given
by the vector sum of bi and b2.
Since the periodic force field of the crystal lattice requires that atoms
must move from one equilibrium position to another, it follows that the
Burgers vector must always connect one equilibrium lattice position with
another. Therefore, the crystal structure will determine the possible
Burgers vectors. A dislocation with a Burgers vector equal to one lattice
.spacing is said to be a dislocation of unit strength. Because of energy
164 Metallurgical Fundamentals [Chap. 6
considerations dislocations with strengths larger than unity are generally
unstable and dissociate into two or more dislocations of lower strength.
The criterion for deciding whether or not dissociation will occur is based ^
on the fact that the strain energy of a dislocation is proportional to the
square of its Burgers vector. Therefore, the dissociation reaction
bi— ^ b2 + bs will occur when br^ > hi^ + 63^, but not if 61" < 62^ + 63^.
Dislocations with strengths less than unity are possible in closepacked
lattices where the equilibrium positions are not the edges of the structure
cell. A Burgers vector is specified by giving its components along the
axes of the crystallographic structure cell. Thus, the Burgers vector for
slip in a cubic lattice from a cube corner to the center of one face has the
components an/2, ao/2, 0. The Burgers vector is [ao/2, ao/2, 0], or, as
generally written, b = (ao/2) [110]. The strength of a dislocation
with Burgers vector aaluvw] is \h\ — ao[w^ + w^ + w'^Y'. For example,
the magnitude of the Burgers vector given above is \h\ = ao/\/2.
A dislocation of unit strength, or unit dislocation, has a minimum energy
when its Burgers vector is parallel to a direction of closest atomic packing
in the lattice. This agrees with the experimental observation that
crystals almost always slip in the closepacked directions. A unit dis
location of this type is also said to be a perfect dislocation because transla
tion equal to one Burgers vector produces an identity translation. For a
perfect dislocation there is perfect alignment of atom planes above and
below the slip plane within the dislocation loop. A unit dislocation par
allel to the slip direction cannot dissociate further unless it becomes an
imperfect dislocation, where a translation of one Burgers vector does not
result in an identity translation. A stacking fault is produced by the
dissociation of a unit dislocation into two imperfect dislocations. For a
stacking fault to be stable, the decrease in energy due to dissociation
must be greater than the increase in interfacial energy of the faulted
region.
64. Dislocations in the Facecentered Cubic Lattice
Shp occurs in the fee lattice on the {111} plane in the (110) direction.
The shortest lattice vector is (ao/2) [110], which connects an atom at a
cube corner with a neighboring atom at the center of a cube face. The
Burgers vector is therefore (ao/2) [110].
However, consideration of the atomic arrangement on the {111} slip
plane shows that slip will not take place so simply. Figure 65 represents
the atomic packing on a closepacked (111) plane. It has already been
shown that the {111} planes are stacked in a sequence ABC ABC • • .
The vector bi = (ao/2) [lOl] defines one of the observed slip directions.
1 F. C. Frank, Physica, vol. 15, p. 131. 1949.
)ec,
64]
Dislocation Theory 165
However, if the atoms are considered as hard spheres, ^ it is easier for an
atom on a type B plane to move along a zigzag path b2 + bg in the valleys
instead of moving over the hump that lies in the path of the vector 61.
The dislocation reaction is given by
bi— > b2 + bs
"[10Tj.f[2TT) + '112]
To check this reaction, the summa
tion of the X, y, z components of the
righthand side of the equation must
add up to the x, y, z components of
the original dislocation.
X component
y component
z component
M = % + M
¥2
76
— 2,
Fig. 65. Slip in a closepacked (111)
plane in an fee lattice. {After A. H.
CottreU, "Dislocations and Plastic Flow
in Crystals," p. 73, Oxford University
Press, New York, 1953.)
The above reaction is energetically
favorable since there is a decrease in
strain energy proportional to the
change ao~/2 —> air/3.
Slip by this twostage process creates a stacking fault ABC A'C ABC in
the stacking sequence. As Fig. 66 shows, the dislocation with Burgers
vector bi has been dissociated into two partial dislocations bo and bs.
This dislocation reaction was suggested by Heidenreich and Shockley,
and therefore this dislocation arrangement is often known as Shockley
partials, since the dislocations are imperfect ones which do not produce
complete lattice translations. Figure 66 represents the situation looking
down on (111) along [111]. AB represents the perfect dislocation line
having the full slip vector bi. This dissociates according to the above
reaction into partial dislocationswith Burgers vectors b2 and bs. The
combination of the two partials AC and AD is known as an extended dis
location. The region between them is a stacking fault representing a part
of the crystal which has undergone slip intermediate between full slip
and no slip. Because b2 and bs are at a 60° angle, there will be a repulsive
force between them (Sec. 69). However, the surface tension of the
stacking fault tends to pull them together. The partial dislocations will
settle at an equilibrium separation determined primarily by the stacking
fault energy. As was discussed in Sec. 410, the stackingfault energy
1 F. C. Thompson and W. E. W. Millington, J. Iron Steel Inst. (London), vol. 109,
p. 67, 1924; C. H. Mathewson, Trans. AIME, vol. 32, p. 38, 1944.
2 R. D. Heidenreich and W. Shockley, "Report on Strength of Solids," p. 37,
Physical Society, London, 1948.
166
Metallurgical Fundamentals
[Chap. 6
can vary considerably for different fee metals and alloys and this in turn
can have an important influence on their deformation behavior.
A characteristic of the fee lattice is that any Burgers vector is common
to two shp planes. This presents the possibility that screw dislocations,
which have no fixed glide plane, may surmount obstacles by gliding onto
another slip plane having a common slip direction. This is the process
of cross slip. However, in order to do this, the extended dislocations
Extended dislocation
C
to =
[121]
■foi]
3 = ^[2Tl]
Fully slipped
No slip
>^1=^[10T]
Fig. 66. Dissociation of a dislocation into two partial dislocations.
must first recombine into perfect dislocations since an extended disloca
tion cannot glide on any plane except the plane of the fault. Figure 426
shows that this requires the formation of a constriction in the stacking
fault ribbon. The greater the width of the stacking fault, or the lower
the stacking fault energy, the more difficult it is to produce constrictions
in the stacking faults. This may explain why cross slip is quite prevalent
in aluminum, which has a very narrow stackingfault ribbon, while it is
difficult in copper, which has a wide stackingfault ribbon.
These ideas are borne out by electronmicroscope transmission studies
of dislocation networks in thin foils. ^ Stacking faults can be readily
detected in these thin films. The nature of the dislocation network in fee
metals changes with the stackingfault energy. Austenitic stainless steel,
with a stackingfault energy around 13 ergs/cm^, shows dislocation net
works only along slip planes, even for large deformations. Gold, copper,
and nickel, where the energy is about 30, 40, andSOergs/cm^, respectively,
^ Hirsch, op. cit.
Sec. 64] Dislocation Theory 167
show the dislocations arranged in complex threedimensional networks at
low strains. This changes into poorly developed subboundaries at higher
deformations. Aluminum, with a stackingfault energy of 200 ergs/cm,
shows almost perfect subboundaries. This picture of a graded transition
in the way the dislocations are arranged is in agreement with the intiuence
of the stackingfault energy on the
ability of a metal to undergo cross
slip. Cross slip is very difficult in
stainless steel, even at high strains,
so that the dislocations are confined
to the slip planes. In gold, copper,
and nickel, cross slip is possible, but
probably only at highly stressed '"'s 67. A Frank partial dislocation or
rj^, n 1 J* sessile dislocation. {After A. H. Cottrell,
regions, i here! ore, cross slip oi r, , ,■ i vi ,■ jpi ■ n
° ' ^ Dislocations and rtastic blow in Crys
SCrew dislocations occurs, and at ^^/^  ^ 75^ Oxford University Press, New
high strains they try to form low York, 1953.)
angle boundary networks to lower
their strain energy. In aluminum, cross slip is very prevalent, and screw
dislocations can easily arrange themselves into a network of lowangle
boundaries.
Frank' pointed out that another type of partial dislocation can exist in
the fee lattice. Figure 67 illustrates a set of (111) planes viewed from
the edge. The center part of the middle A plane is missing. An edge
dislocation is formed in this region with a Burgers vector (ao/3)[lll].
This is called a Frank partial dislocation. Its Burgers vector is per
pendicular to the central stacking fault. Since glide must be restricted
to the plane of the stacking fault and the Burgers vector is normal to
this plane, the Frank partial dislocation cannot move by glide. For this
reason it is called a sessile dislocation. A sessile dislocation can move only
by the diffusion of atoms or vacancies to or from the fault, i.e., by the
process of climb. Because climb is not a likely process at ordinary
temperatures, sessile dislocations provide obstacles to the movement of
other dislocations. Dislocations which glide freely over the slip plane,
such as perfect dislocations or Shockley partials, are called glissile. A
method by which a missing row of atoms can be created in the (111)
plane is by the condensation of a disk of vacancies on that plane. Evi
dence for the collapse of disks of vacancies in aluminum has been obtained
by transmission electron microscopy. 
Sessile dislocations are produced in the fee lattice by the glide of dis
locations on intersecting (111) planes. These sessile dislocations are
1 F. C. Frank, Proc. Phys. Soc. (London), vol. 62A, p. 202, 1949.
' P. B. Hirsch, J. Silcox, R. E. Smallman, and K. H. Westmacott, Phil. Mag., vol. 3,
p. 897, 1958.
168 Metallurgical Fundamentals
[Ch
ap.
known as CottrellLomer harriers. They are an important element in the
mechanism of the strain hardening of metals. Lomer^ pointed out that
dislocations moving on intersecting slip planes will attract and combine
if their Burgers vectors have suitable orientations. Figure 68 illustrates
two dislocations moving on the slip planes of an fee lattice. Dislocation
A is moving in a (111) plane with a Burgers vector (ao/2)[101]. Disloca
tion B glides in a (111) plane with a Burgers vector (ao/2)[011]. These
dislocations attract each other and move
toward the intersection point 0, which is
the intersection of the two Burgers vectors
along the direction [110]. At this point
// the two dislocations react according to
Lomer's reaction
f [101]+ I [Oil]
ao
[110]
Fig. 68. Dislocation reaction
leading to CottrellLomer bar
riers. {After A. H. Cottrell,
"Dislocations and Plastic Flow
in Crystals," p. 171, Oxford
University Press, New York,
1953.)
to form a new dislocation of reduced energy.
Since all three dislocations must be parallel
to the line of intersection of the slip plane,
[lIO], the edge dislocation formed by
Lomer's reaction has a slip plane (001).
The plane (001) contains both the Burgers
vector [110] and the line [lIO]. Since (001)
is not a common slip plane in the fee lattice,
the dislocation formed from Lomer's reac
tion should not glide freely. However, it is not a true sessile disloca
tion, in the sense of the Frank partial, because it is not d,n imperfect
dislocation.
CottrelP showed that the product of Lomer's reaction could be made
truly immobile by the following dislocation reaction :
I [110] ^ ^ [112] + ^ [112] + ^ [110]
The products of this dislocation reaction are imperfect edge dislocations
which form the boundaries of stacking faults. The dislocation (ao/6)[112]
is a Shockley partial which glides in the (111) plane. It is repelled from
the line and forms a stacking fault bounded by two [110] lines, the line
and the line of the dislocation. In a similar way, the dislocation
(ao/6)[112] glides in the (111) plane and forms a stacking fault bounded
by the line and the line of the dislocation. The third dislocation with
Burgers vector (ao/6)[110] hes along the line where the two stacking
» W. M. Lomer, Phil. Mag., vol. 42, p. 1327, 1951.
2 A. H. Cottrell, Phil. Mag., vol. 43, p. 645, 1952.
Sec. 66] Dislocation Theory 169
faults join. This combination of three dislocations produced by the
CottrellLomer reaction forms an isosceles triangle which is locked
rigidly in place and cannot glide. Therefore, CottrellLomer locking
provides an effective barrier to slip. Studies by transmission electron
microscopy of dislocation interaction in thin foils have confirmed the
existence of interaction that is in agreement with the model of Cottrell
Lomer locking. 1
CottrellLomer barriers can be overcome at high stresses and/or
temperatures. A mathematical analysis of the stress required to break
down a barrier either by slip on the (001) plane or by dissociation into
the dislocations from which it was formed has been given by Stroh.^
However, it has been shown'' that for the important case of screw disloca
tions piled up at CottrellLomer barriers the screw dislocations can
generally escape the pileup by cross slip before the stress is high enough
to collapse the barrier.
65. Dislocations in the Hexagonal Closepacked Lattice
The basal plane of the hep lattice is a closepacked plane with the
stacking sequence ABABAB • • • . Slip occurs on the basal plane
(0001) in the <1120> direction (Fig. 43). The smallest unit vector for the
hep structure has a length ao and lies in the closepacked (1120) direction.
Therefore, the Burgers vector is ao[1120]. Dislocations in the basal plane
can reduce their energy by dissociating into Shockley partials according
to the reaction
ao[1120]^ ao[10lO] + ao[OlTO]
The stacking fault produced by this reaction lies in the basal plane, and
the extended dislocation which forms it is confined to glide in this plane.
66. Dislocations in the Bodycentered Cubic Lattice
Shp occurs in the (111) direction in the bcc lattice. The shortest
lattice vector extends from an atom corner to the atom at the center
of the unit cube. Therefore, the Burgers vector is (ao/2)[lll]. It will
be recalled that shp lines in iron have been found to occur on {110},
{112}, and {123}, although in other bcc metals slip appears to occur
predominantly on the {110} planes.
1 M. J. Whelan, Proc. Roy. Soc. (London), vol. 249A, p. 114, 1958; all possible dis
location reactions in the fee lattice have been worked out by J. P. Hirth, J. Appl.
Phys., vol. 32, pp. 700706, 1961.
2 A. N. Stroh, Phil. Mag., vol. 1, ser. 8, p. 489, 1956.
3 A. Seeger, J. Diehl, S. Mader, and R. Rebstock, Phil. Mag., vol. 2, p. 323, 1957.
170 Metallurgical Fundamentals [Chap. 6
Dislocation reactions have not been studied so extensively in the bcc
lattice as in the fee lattice. Cottrell^ has suggested that a perfect dis
location in a (112) plane can dissociate according to the reaction
f [111] ^ 1° [112] + 1° [111]
The dislocation (ao/3)[112] is a pure edge dislocation since its Burgers
vector lies perpendicular to the slip plane. It is also an imperfect sessile
dislocation that forms the boundary of a stacking fault in the (112) planes.
The dislocation (ao/6)[lll] is an imperfect glissile dislocation similar to
the Shockley partial of the fee lattice. However, because [111] is the
line of intersection of three planes of the type {112}, this dislocation can
glide out of the plane of the stacking fault too easily to be part of a true
extended dislocation. A dislocation in the (112) plane may also lower its
energy by dissociating according to the reaction
f [lll]>°[lll]+f [111]
As discussed above, both the partial dislocations formed by this reaction
are pure screw, and because of the geometry of the situation, they are not
completely confined to the (112) slip plane. An analysis^ of the atomic
positions giving rise to stacking faults on {112} planes shows that there
are two types which may result. While the existence of stacking faults
in the bcc lattice has been demonstrated by Xray diffraction, detailed
studies of the dislocation reactions discussed in this paragraph have not
yet been made.
CottrelP has suggested another dislocation reaction, which appears to
lead to the formation of immobile dislocations in the bcc lattice. This
dislocation reaction may be important to the brittle fracture of bcc metals.
Consider Fig. 69a. Dislocation A, with Burgers vector (ao/2)[lll],
is gliding in the (101) plane. Dislocation B, with Burgers vector
(ao/2)[lll], is gliding in the intersecting slip plane (101). The two
dislocations come together and react to lower their strain energy by the
reaction
f [ni]f[lll]^ao[001]
The product of this reaction is a pure edge dislocation which lies on the
(001) plane. Since this is not a common slip plane in the bcc lattice,
1 A. H. Cottrell, "Dislocations and Plastic Flow in Crystals," Oxford University
Press, New York, 1953.
2 J. M. Silcock, Acta Met., vol. 7, p. 359, 1959.
3 A. H. Cottrell, Trans. Met. Soc. AIMS, vol. 212, p. 192, 1958.
Sec. 67]
Dislocation Theory 171
the dislocation is immobile. However, the (001) plane is the cleavage
plane along which brittle fracture occurs. Cottrell suggests that the
formation of a dislocation on the cleavage plane by slip on intersecting
{110} planes is equivalent to introducing a crack one lattice spacing thick
(Fig. 695). This crack can then grow by additional dislocations gliding
over the {110} planes. While this particular dislocation reaction has not
Fi9. 69. Slip on intersecting (110) planes. (.4. H. Cottrell, Trans. AIME, vol. 212,
p. 196, 1958.)
been established by experiment in bcc metals, it has been found to oper
ate in cubic ionic crystals such as LiF and MgO.
67. Stress Field of a Dislocation
A dislocation is surrounded by an elastic stress field that produces
forces on other dislocations and results in interaction between dislocations
and solute atoms. For the case of a perfect dislocation a good approxi
mation of the stress field can be obtained from the mathematical theory
of elasticity for continuous media. However, the equations obtained
are not valid close to the core of the dislocation line. The equations
given below apply to straight edge and screw dislocations in an isotropic
crystal. 1 The stress around a straight dislocation will be a good approxi
mation to that around a curved dislocation at distances that are small
compared with the radius of curvature. Appreciably greater complexity
results from the consideration of a crystal with anisotropic elastic
constants.^
Figure 610 represents the cross section of a cylindrical piece of elastic
material containing an edge dislocation running through point parallel
to the z axis (normal to the plane of the figure). The original undistorted
cylinder without a dislocation is shown by the dashed line. The dis
1 For derivations see F. R. N. Nabarro, Advances in Phys., vol. 1, no. 3, pp. 271395,
1952; W. T. Read, Jr., "Dislocations in Crystals," pp. 114123, McGrawHill Book
Company, Inc., New York, 1953.
"^ J. D. Eshelby, W. T. Read, and W. Shockley, Acta Met., vol. 1, pp. 351359, 1953,
172 Metallurgical Fundamentals
[Chap. 6
location was produced by making
a radial cut along the plane ?/ =
(line OA), sliding the cut surfaces
along each other the distance A A',
and joining them back together
again. This sequence of operations^
produces a positive edge dislocation
running along the z axis with a strain
field identical with that around a
dislocation model such as that of
Fig. 48. Since the dislocation line
is parallel to the z axis, strains in
that direction are zero and the prob
lem can be treated as one in plane
strain.
For the case of a straight edge
dislocation in an elastically isotropic
material the stresses, in terms of
three orthogonal coordinate axes, are given by the following equations.
The notation is the same as that used in Chaps. 1 and 2.
Fig. 610. Deformation of a circle con
taining an edge dislocation. The un
strained circle is shown by a dashed line.
The solid line represents the circle after
the dislocation has been introduced.
To
To
by(3x + ij^)
by(x'  y^)
where
To =
G
27r(l  v)
bx{x^ — y'^)
(x^ \ y'^'Y
.. =
Txy — To
Txz Ty,
For polar coordinates, the equations are
—Toh sin Q
CTr — (Te —
r
h cos d
TrB — TBt — To
(61)
(62)
(63)
(64)
(65)
(66)
(67)
ar acts in the radial direction, while ae acts in a plane perpendicular to r.
Note that the stresses vary inversely with distance from the dislocation
1 It is interesting that this problem was analyzed by Volterra in 1907, long before
the concept of dislocations was originated. The mathematical details may be found
in A. E. H. Love, "A Treatise on the Mathematical Theory of Elasticity," pp. 221^
228, Cambridge University Press, New York, 1934.
Sec. 67] Dislocation Theory 173
line. Since the stress becomes infinite at r = 0, a small cylindrical
region r = ro around the dislocation line must be excluded from the
analysis.
A straight screw dislocation in an isotropic medium has complete
cylindrical symmetry. For a rectangularcoordinate system only two
components of stress are not equal to zero.
 ^ a;
Since there is no extra half plane of atoms in a screw dislocation, there
are no tensile or compressive normal stresses. The stress field is simply
one of shear. The radial symmetry of this stress field is apparent when
the shear stress is expressed in a polarcoordinate system.
r.. = ^^ (610)
The strain field around an edge dislocation in a silicon crystal has been
observed' by means of polarized infrared radiation. The variation in
intensity is in agreement with what would be expected from the equa
tions for a stress field around an edge dislocation in an isotropic medium.
The strain energy involved in the formation of an edge dislocation
can be estimated from the work involved in displacing the cut OA in
Fig. 610 a distance h along the slip plane.
1 /''■' 1 /"'■' dr
U = n\ Trebdr =  rob" cos ^— (611)
2 Jro 2 Jro r
But cos 6=1 along the slip plane y = 0, so that the strain energy is
given by
U = . ,^" , In ^ (612)
47r(l  v) ro
In the same way, the strain energy of a screw dislocation is given by
U = ^ \ Te^h dr = ^\n ' (613)
2 Jro 47r ro
Note that, in accordance with our assumption up to this point, the strain
energy per unit length of dislocation is proportional to Gh'^. This strain
energy corresponds to about 10 ev for each atom plane threaded by an
edge dislocation (Prob. 69). The total energy of a crystal containing
1 W. L. Bond and J. Andrus, Phys. Rev., vol. 101, p. 1211, 1956.
174 Metallurgical Fundamentals
[Chap. 6
many dislocation lines is the si^t^ f the strain energies of the individiml
dislocations, plus terms expressing the interactions of the stress fields of
the dislocations, plus a term describing the internal stresses developed
by the external forces.
68. Forces on Dislocations
When an external force of sufficient magnitude is applied to a crystal,
the dislocations move and produce slip. Thus, there is a force acting on
a dislocation line which tends to drive it
forward. Figure 611 shows a dislocation
line moving in the direction of its Burgers
vector under the influence of a uniform
shear stress t. An element of the disloca
tion line ds is displaced in the direction
of slip normal to ds by an amount dl.
The area swept out by the line element
is then ds dl. This corresponds to an
average displacement of the crystal above
the slip plane to the crystal below the
slip plane of an amount ds dlb/A, where
A is the area of the slip plane. The work
done by the shear stress acting in the slip
plane is dW = TA{ds dlh)/A. This corresponds to a force dW/dl act
ing on the element ds in the direction of its normal. Therefore, the force
per unit length acting on the dislocation line is
Fig. 611. Force acting on a dis
location line.
F = Th
(614)
This force is normal to the dislocation line at every point along its length
and is directed toward the unslipped part of the glide plane.
Because the strain energy of a dislocation line is proportional to its
length, work must be performed to increase its length. Therefore, it is
convenient to consider that a dislocation possesses a line tension which
attempts to minimize its energy by shortening its length. The line ten
sion has the units of energy per unit length and is analogous to the surface
tension of a liquid. For a curved dislocation line, the line tension pro
duces a restoring force which tends to straighten it out. The magnitude
of this force is T/R, where r is the line tension and R is the radius of
curvature of the bent dislocation line. The direction of this force is
perpendicular to the dislocation line and toward the center of curvature.
Because of line tension, a dislocation line can have an equilibrium curva
ture only if it is acted on by a shear stress. The equilibrium condition
Sec. 69] Dislocation Theory 175
for this to occur is
Therefore, the shear stress needed to maintain a dislocation line in a
radius of curvature R is
r = i^ (615)
Orowan^ has pointed out that the determination of this stress bears an
analogy with the problem of blowing a bubble from a nozzle submerged
in a liquid. The line tension will vary from point to point along a dis
location line. Stroh^ has shown that Eq. (613) provides a good approxi
mation of the line tension. An approximation often used is F ~ 0.5Gb^.
This is obtained from Eq. (613) when typical values ri = 1,000 A and
ro = 2 A are used.
69. Forces between Dislocations
Dislocations of opposite sign on the same slip plane will attract each
other, run together, and annihilate each other. This can be seen readily
for the case of an edge dislocation (Fig. 48), where the superposition of
a positive and negative dislocation on the same slip plane would elimi
nate the extra plane of atoms and therefore the dislocation would dis
appear. Conversely, dislocations of like sign on the same slip plane will
repel each other.
The simplest situation to consider is the force between two parallel
screw dislocations. Since the stress field of a screw dislocation is radially
symmetrical, the force between them is a central force which depends
only on the distance that they are apart.
Fr = re.h = ^ (616)
The force is attractive for dislocations of opposite sign (antiparallel
screws) and repulsive for dislocations of the same sign (parallel screws).
Consider now the forces between two parallel edge dislocations with
the same Burgers vectors. Referring to Fig. 610, the edge dislocations
are at P and Q, parallel to the z axis, with their Burgers vectors along
the X axis. The force between them is not a central force, and so it is
necessary to consider both a radial and a tangential component. The
^ E. Orovvan, "Dislocations in Metals," pp. 99102, American Institute of Mining
and Metallurgical Engineers, New York, 1953.
2 A. N. Stroh, Proc. Phys. Soc. (London), vol. 67B, p. 427, 1954.
176 Metallurgical Fundamentals
force per unit length is given by^
Gb' 1
Gh'
sin 20
[Chap. 6
(617)
27r(l  v)r 27r(l  v) r
Because edge dislocations are mainly confined to the slip plane, the force.
0.3
Fig. 612. Graphical representation of Eq. (618). Solid curve A is for two edge dis
locations of same sign. Dashed curve B is for two unlike edge dislocations. {After
A. H. Cottrell, "Dislocations and Plastic Flow in Crystals," p. 48, Oxford University
Press, New York, 1953.)
component along the x direction, which is the slip direction, is of most
interest.
F:c = Fr cos e  Fe sin 6
_ Gh^x(x^ — y'^)
~ 27r(l  p)(x^^ + yT
(618)
Figure 612 is a plot of the variation of F^ with distance x, where x is
expressed in units of y. Curve A is for dislocations of the same sign;
curve B is for dislocations of opposite sign. Note that dislocations of
the same sign repel each other when x > y (6 < 45°) and attract each
other when x < y {d > 45°). The reverse is true for dislocations of
1 A. H. Cottrell, "Dislocations and Plastic Flow in Crystals," p. 46, Oxford Uni
versity Press, New York, 1953.
Sec. 610] Dislocation Theory 177
opposite sign. Fx is zero at x = and x = y. The situation a; = 0,
where the edge dislocations lie vertically above one another, is a con
dition of equilibrium. Thus, theory predicts that a vertical array of
edge dislocations of the same sign is in stable equilibrium. This is the
arrangement of dislocations that exists in a lowangle grain boundary
of the tilt variety.
The situation of two parallel dislocations with different Burgers vec
tors can be rationalized by considering their relative energies.' This
represents the situation of dislocations on two intersecting slip planes.
In general there will be no stable position, as for the previous case. The
dislocations either will try to come together or will move far apart. Con
sider two parallel dislocations bi and b2, which may or may not attract
and combine into bs. The two dislocations will attract if 63^ < 61^ \ 62^
and will repel if 63^ > bi^ + 62^ Expressed another way, the disloca
tions will attract if the angle between their Burgers vectors is greater
than 90°. They will repel if it is less than 90°.
A free surface exerts a force of attraction on a dislocation, since escape
from the crystal at the surface would reduce its strain energy. Koehler^
has shown that this force is approximately equal to the force which would
be exerted in an infinite solid between the dislocation and one of opposite
sign located at the position of its image on the other side of the surface.
This image force is equal to
F = , ,f^' ,  (619)
47r(l — v) r
for an edge dislocation. However, it should be noted that metal surfaces
are often covered with thin oxide films. A dislocation approaching a
surface with a coating of an elastically harder material will encounter a
repulsive rather than an attractive image force.
610. Dislocation Climb
An edge dislocation can glide only in the plane containing the disloca
tion line and its Burgers vector (the slip direction). To move an edge
dislocation in a direction perpendicular to the slip plane requires the
process of climb. The motion of a screw dislocation always involves
glide, so that it is not involved with climb. Climb requires mass trans
port by diffusion, and therefore it is a thermally activated process. By
convention, the positive direction of climb is the direction in which atoms
are taken away from the extra half plane of atoms in an edge dislocation
so that this extra half plane moves up one atomic layer. The usual way
1 Read, op. cit., p. 131.
2 J. S. Koehler, Phys. Rev., vol. 60, p. 397, 1941.
178 Metallurgical Fundamentals [Chap. 6
for this to occur is by a vacancy diffusing to the dislocation and the
extra atom moving into the vacant lattice site (Fig. 613). It is also
possible, but not energetically favorable, for the atom to break loose from
the extra half plane and become an interstitial atom. To produce nega
tive climb, atoms must be added to the extra half plane of atoms. This
can occur by atoms from the surrounding lattice joining the extra half
plane, which creates vacancies, or,
less probably, by an interstitial atom
diffusing to the dislocation.
• • V^ • • Dislocation climb is necessary to
• • ' • • bring about the vertical alignment of
• • • « edge dislocations on slip planes that
(^) ,.^ produces lowangle grain boundaries
by the process of polygonization.
Fig. 61 3. (a) Diffusion of vacancy to t?., i^ i . t ■ u j. j
, J. , \. ,,, ,. , ,. ,f , JtLitchpit techniques on bent and
edge dislocation; (o) dislocation climbs ^ .
up one lattice spacing. annealed crystals have amply dem
onstrated the existence of this phe
nomenon. Dislocation climb is also a very important factor in the creep
of metals, where the activation energy for steadystate creep is equal to
the activation energy for selfdiffusion in pure metals. Since selfdif
fusion occurs by the movement of vacancies, this implies that dislocation
climb is involved in creep.
61 1 . Jogs in Dislocations
There is no requirement that a dislocation must be confined to a single
plane. When a dislocation moves from one slip plane to another, it cre
ates a step, or jog, in the dislocation line. Jogs can be produced by the
intersection of dislocations, as was shown earlier, in Fig. 429, or a jog
can be produced during climb owing to the failure of climb to occur along
the entire length of the extra half plane of atoms.
The intersection of two edge dislocations is illustrated in Fig. 614.
An edge dislocation XY with a Burgers vector bi is moving on plane Pxy.
It cuts through dislocation AD, with Burgers vector h, lying on plane
Pad. The intersection produces a jog PP' in dislocation AD. The
resulting jog is parallel to bi, but it has a Burgers vector b since it is
part of the dislocation line APP'D. The length of the jog will be equal
to the length of the Burgers vector bi. It can be seen that the jog result
ing from the intersection of two edge dislocations has an edge orientation,
and therefore it can readily glide with the rest of the dislocation. Hence,
the formation of jogs in edge dislocations will not impede their motion.
However, it requires energy to cut a dislocation because the formation
Sec. 612]
Dislocation Theory 179
/^
Y
X
^
bx
/I ^
Pad
^ \ ^
.^
PxY^^
[a)
^^
of a jog increases its length. The energy of a jog will be about 0.5G'6^,
since the average line tension is 0.5Gb^ and the jog has a length 6,.
Figure 429 illustrates the intersection of two screw dislocations. As is
the general rule when jogs are
formed by dislocation intersection,
the jogs are perpendicular to the
slip planes in which the dislocations
are moving. It can be seen that
the jogs formed by the intersection
of two screw dislocations have an
edge orientation because they lie
perpendicular to the Burgers vector
of the screw dislocations. Since an
edge dislocation can move easily
only in the plane containing its line
and its Burgers vector, the jog can
move only along the screw axis of
the dislocation. Therefore, so long
as the jog is present on the screw
dislocation, it cannot move in a
direction normal to the screw axis
except by the process of climb.
Hence, it is more difficult to move
screw dislocations through an inter
secting forest of dislocations than it
is to move edge dislocations through
an intersecting array. This is sub
stantiated by the observation^ that
slip bands in aluminum advance
more slowly when viewed in a di
rection perpendicular to the slip
direction than when viewed along
the slip direction. For the intersection of mixed dislocations, part edge
and part screw, the jog can move sidewise by glide as the dislocation
moves through the lattice.
Fig. 614. Intersection of two edge dis
locations. (W. T. Read, Jr., "Disloca
tions in Crystals," McGrawHill Book
Company, Inc., New York, 1953.)
61 2. Dislocation and Vacancy Interaction
There is a growing amount of evidence that point defects, mainly
vacancies, are produced during plastic deformation. Most of the experi
' N. K. Chen and R. B. Pond, Trans. AIME, vol. 194, pp. 10851092, 1952.
180 Metallurgical Fundamentals [Chap. 6
mental evidence' is based on deformation at low temperature (so as to
suppress the mobility of vacancies) followed by the measurement of elec
trical resistivity and mechanical strength before and after annealing treat
ments. It is found that about half the increased resistivity due to cold
work anneals out over welldefined temperature ranges and with acti
vation energies which generally agree with the temperatures and acti
vation energies observed for the annealing of quenched and irradiated
samples. Moreover, the changes in resistivity are accomplished with
little change in mechanical strength, indicating that dislocations are not
responsible for the resistivity changes. The generation of point defects
due to deformation has been demonstrated in ionic crystals by measure
ments of conductivity and density and by the observation of color centers.
Jogs in dislocation lines can act as sources and sinks for point defects.
Because of the reentrant corner at a jog, it is a favorable center for the
absorption and annihilation of vacancies. It is also generally considered
that vacancies can be generated at jogs. The usual mechanism^ involves
the jogs formed by the intersection of screw dislocations. As was pointed
out in the previous section, motion of a screw dislocation containing jogs
in a direction normal to its axis can occur only by climb. As the jog
climbs, it generates vacancies. However, two points of doubt have been
raised about this mechanism. Friedel^ has pointed out that there is no
reason why a jog should not glide along a screw dislocation without pro
ducing vacancies so long as it can shortly attach itself to an edge com
ponent of the dislocation line. Cottrell^ has shown that the jogs formed
by intersecting screw dislocations will generally produce interstitials, not
vacancies. However, annealing experiments show that vacancies rather
than interstitials are the predominant point defect in coldworked metals.
Other mechanisms for the generation of vacancies by jogs on dislocations
have been proposed by Friedel, Mott, and Cottrell.^ While the exact
details for the mechanism of vacancy formation during cold work have
not been established, there is little question that jog formation due to
the intersection of dislocations is involved.
An attractive force exists between vacancies and dislocations. There
1 For reviews of this subject see T. Broom, Advances in Phys., vol. 3, pp. 2683,
1954, and "Symposium on Vacancies and Other Point Defects in Metals and Alloys,"
Institute of Metals, London, 1958.
2 F. Seitz, Advances in Phys., vol. 1, p. 43, 1952.
3 J. Friedel, Phil. Mag., vol. 46, p. 1165, 1955.
^ A. H. Cottrell, "Dislocations and Mechanical Properties of Crystals," pp. 509
512, John Wiley & Sons, Inc., New York, 1957.
5 J. Friedel, "Les Dislocations," GauthierVillars & Cie, Paris, 1956; N. F. Mott,
"Dislocations and Mechanical Properties of Crystals," pp. 469471, John Wiley &
Sons, Inc., New York, 1957; A. H. Cottrell, "Vacancies and Other Point Defects in
Metals and Alloys," pp. 2829, Institute of Metals, London, 1958.
Sec. 61 3] Dislocation Theory 181
fore, vacancies should be able to form atmospheres around dislocations
in the same way as solute atoms. Vacancies may also interact with each
other to form vacancy pairs (divacancies), and there is some evidence to
support the hypothesis that they collect into larger groups or clusters.
61 3. Dislocation — Foreignatom Interaction
The presence of a large foreign atom produces a dilation of the matrix.
An oversized atom will be attracted to the tension region and repelled
from the compression region of an edge dislocation. The segregation of
solute atoms to dislocations lowers the strain energy of the system. For
simplicity, it is assumed that the solute atom produces a symmetrical
hydrostatic distortion of the matrix. If the solute atom occupies a vol
ume AV greater than the volume of the matrix atom it replaces, the energy
of interaction between the local stress field of the dislocation and the
foreign atom will be
Ui = a^AV (620)
where o„, = — Is{(Tx + c^ + (7^) is the hydrostatic component of the stress
field. The volume change is given by
AV = %wea^ (621)
where a is the radius of the solvent atom and e = (a' — a) /a is the strain
produced by introducing a solute atom of radius a'. When the solute
atom is located at a point given by the polar coordinate r, 6 from an edge
dislocation, the interaction energy is given by^
U, = ^^^^ = 4Gbea^'^ (622)
r r
The force between an edge dislocation and a solute atom is not a central
force. The radial and tangential components are given by
When the solute atom produces an unequal distortion of the matrix
lattice in different directions, solute atoms can interact with the shear
component of the stress field as well as the hydrostatic component.
Under these conditions interaction occurs between solute atoms and both
screw and edge dislocations. For the case of carbon and nitrogen atoms
in iron the tetragonal symmetry around the interstitial sites leads to a
shear component of the stress field. In fee alloys the dissociation of dis
1 B. A. Bilby, Proc. Phys. Soc. (London), vol. 63A, p. 191, 1950.
182 Metallurgical Fundamentals [Chap. 6
locations into partial dislocations produces two elastically bound dis
locations with a substantial edge component.
Cottrell and Bilbj^ have shown that in time t the number of solute
atoms, n{t), that migrate to a unit length of dislocation line from a solu
tion containing initially no solute atoms per unit volume is
^(0 = 3(^y''(^^y'no (624)
where A = interaction parameter of Eq. (622)
D = diffusion coefficient of solute atoms at temperature T
In the derivation of this equation the dislocation line serves as a solute
atom sink which captures any passing atom but does not obstruct the
entry of other atoms. This concept is valid during the early stages of
strain aging, where the f'^ relationship is found to hold. However,
toward the later stages of strain aging the sites on the dislocation line
become saturated, and the assumption that it acts like a sink can no
longer be valid. Now the probability of atoms leaving the center equals
the probability of atoms flowing in, and a steadystate concentration
gradient develops. The steadystate distribution of solute atoms around
the dislocation is referred to as an atmosphere. The local concentration
c is related to the average concentration co by the relationship
c = Co exp j~ (625)
It has been suggested^ that solute atoms can diffuse along dislocations
until they meet a barrier. If the interaction between the solute atoms is
strong, a fine precipitate can be formed. In this way the dislocation lines
are freed to act as sinks for a longer period of time, and the fi^ relation
ship will remain valid until all dislocation lines have been saturated with
solute atoms.
When the concentration of solute atoms around the dislocation becomes
high enough, the atmosphere will condense into a single line of solute
atoms parallel to the dislocation line at the position of maximum binding
about two atomic spacings below the core of a positive edge dislocation.
The breakaway stress required to pull a dislocation line away from a line
of solute atoms at 0°K is
6Vo^
(626)
where A is given by Eq. (622) and ro « 2 X 10"^ cm is the distance from
the dislocation core to the site of the line of solute atoms. When the
1 B. A. Bilby and G. M. Leak, /. Iron Steel Inst. (London), vol. 184, p. 64, 1956.
Sec. 614] Dislocation Theory 183
dislocation line is pulled free from the field of influence of the solute
atoms, slip can proceed at a stress lower than that given by Eq. (626).
This is the origin of the upper yield point in the stressstrain curve.
When an external force tries to move a dislocation line away from its
atmosphere, the atmosphere exerts a restoring force that tries to pull it
back to its original position. If the speed of the dislocation line is slow,
it may be able to move by dragging the atmosphere along behind it.
According to Cottrell, the maximum velocity at which a dislocation line
can move and still drag its atmosphere with it is
If the dislocation line is moving faster than this velocity, it will be neces
sary for the restoring force to be overcome and the atmosphere is left
behind. Serrations in the stressstrain curve are the result of the dis
location line pulling away from the solute atmosphere and then slowing
down and allowing the atmosphere to interact once again with the
dislocations.
614. Dislocation Sources
The low yield strength of pure crystals leads to the conclusion that
dislocation sources must exist in completely annealed crystals and in
crystals carefully solidified from the melt. The line energy of a disloca
tion is so high as to make it very unlikely that stresses of reasonable
magnitude can create new dislocations in a region of a crystal where no
dislocations exist, even with the assistance of thermal fluctuations. This
results in an important difference between line defects and point defects.
The density of dislocations in thermal equilibrium with a crystal is vanish
ingly small. There is no general relationship between dislocation density
and temperature such as exists with vacancies. Since dislocations are
not affected by thermal fluctuations at temperatures below which recrys
tallization occurs, a metal can have widely different dislocation densities
depending upon processing conditions. Completely annealed material
will contain about 10^ to 10* dislocation lines per square centimeter,
while heavily coldworked metal will have a dislocation density of about
10'^ dislocation lines per square centimeter.
It is generally believed that all metals, with the exception of tiny
whiskers, initially contain an appreciable number of dislocations, pro
duced as the result of the growth of the crystal from the melt or the
vapor phase. Experimental evidence for dislocations in crystals solidi
fied under carefully controlled conditions has been obtained by etchpit
studies and by Xray diffraction methods. For crystals grown by vapor
184 Metallurgical Fundamentals [Chap. 6
deposition it has been shown that nucleation of the soUd phase occurs
around screw dislocations emerging from the surface of the solid substrate.
Ample evidence of the existence of threedimensional dislocation net
works in annealed ionic crystals has been provided by dislocation deco
ration techniques. In annealed metals, dislocation loops have been
observed by transmissionelectron microscopy of thin films. ^ These
loops are believed to originate from the collapse of disks of vacancies
and correspond to prismatic dislocations. There is some evidence to
indicate that these loops can grow and join up to form dislocation net
works in annealed, unworked crystals. There is also some evidence to
suggest that some of the condensed vacancies form voids, which are then
responsible for the formation of dislocations. While there is little doubt
that dislocations exist in annealed or carefully solidified metal, much more
information is needed about the mechanism, by which they are produced
and the way in which they are arranged in the metal.
61 5. Multiplication of Dislocations — FrankRead Source
One of the original stumbling blocks in the development of dislocation
theory was the formulation of a reasonable mechanism by which sources
originally present in the metal could produce new dislocations by the
process of slip. Such a mechanism is required when it is realized that
the surface displacement at a slip band is due to the movement of about
1,000 dislocations over the slip plane. Thus, the number of dislocation
sources initially present in a metal could not account for the observed
slipband spacing and displacement unless there were some way in which
each source could produce large amounts of slip before it became immobi
lized. Moreover, if there were no source generating dislocations, cold
work should decrease, rather than increase, the density of dislocations in
a single crystal. Thus, there must be a method of generating dislocations
or of multiplying the number initially present to produce the high dis
location density found in coldworked metal. The scheme by which dis
locations could be generated from existing dislocations was proposed by
Frank and Read^ and is commonly called a FrankRead source.
Consider a dislocation line DD' lying in a slip plane (Fig. 61 5a). The
plane of the figure is the slip plane. The dislocation line leaves the slip
plane at points D and D', so that it is immobilized at these points. This
could occur if D and D' were nodes where the dislocation in the plane of
the paper intersects dislocations in other slip planes, or the anchoring
could be caused by impurity atoms. If a shear stress t acts in the slip
plane, the dislocation line bulges out and produces slip. For a given
^ Hirsch, Silcox, Smallman, and Westmacott, op. cit.
2 F. C. Frank and W. T. Read, Phys. Rev., vol. 79, pp. 722723, 1950.
Sec. 61 5]
Dislocation Theory 185
stress the dislocation line will assume a certain radius of curvature given
by Eq. (615). The maximum value of shear stress is required when the
dislocation bulge becomes a semicircle so that R has the minimum value
1/2 (Fig. 6156). From the approximation that V ~ Q.bGh'^ and Eq.
Tb
III ,
[a)
Tb
k
[b)
.Tb
(d)
[e)
Fig. 61 5. Schematic representation of the operation of a FrankRead source. {W . T.
Read, Jr., "Dislocations in Crystals,'' McGrawHill Book Company, Inc., New York,
1953.)
(615) it can be readily seen that the stress required to produce this
configuration is
Gh
(628)
where I is the distance DD' between the nodes. When the stress is raised
above this critical value, the dislocation becomes unstable and expands
indefinitely. Figure 61 5c shows the expanded loop, which has started
to double back on itself. In Fig. 61 5rf the dislocation has almost doubled
back on itself, while in Fig. 615e the two parts of the loop have joined
together. This produces a complete loop and reintroduces the original
dislocation line DD' . The loop can continue to expand over the slip
plane with increasing stress. The section DD' will soon straighten out
under the influence of applied stress and line tension, and the Frank
Read source will then be in a position to repeat the process. This process
can be repeated over and over again at a single source, each time pro
186 Metallurgical Fundamentals [Chap. 6
diicing a dislocation loop which produces slip of one Burgers vector along
the slip plane. However, once the source is initiated, it does not con
tinue indefinitely. The back stress produced by the dislocations piling
up along the slip plane opposes the applied stress. When the back stress
Fig. 616. FrankRead source in silicon crystal. [W. C. Dash, in "Dislocations and
Mechanical Properties of Crystals," John Wiley & Sons, Inc., New York, 1957.)
equals the critical stress given by Eq. (628), the source will no longer
operate.
The most dramatic evidence for the existence of a FrankRead source
has been found by Dash^ in silicon crystals decorated with copper. Fig
ure 616 shows a FrankRead source in a silicon crystal as photographed
with infrared light. Evidence has also been found by precipitation tech
niques in aluminum alloys and in ionic crystals and by means of thinfilm
electron microscopy in stainless steel.
616. Dislocation Pileup
Frequent reference has been made to the fact that dislocations pile up
on slip planes at obstacles such as grain boundaries, secondphase parti
cles, and sessile dislocations. The dislocations in the pileup will be
^ W. C. Dash, "Dislocations and Mechanical Properties of Crystals," p. 57, John
Wiley & Sons, Inc., New York, 1957.
Sec. 616] Dislocation Theory 187
tightly packed together near the head of the array and more widely
spaced toward the source (Fig. 617). The distribution of dislocations
of like sign in a pileup along a single slip plane has been studied by
Eshelby, Frank, and Nabarro.^ The number of dislocations that can
Source
/
Fig. 617. Dislocation pileup at an obstacle.
occupy a distance L along the slip plane between the source and the
obstacle is
klTTsL
n =
Gh
(629)
where r,, is the average resolved shear stress in the slip plane and k is a
factor close to unity. For an edge dislocation k = I — v, while for a
screw dislocation A = 1. When the source is located at the center of
a grain of diameter D, the number of dislocations in the pileup is given by
n = ^^ (630)
The factor 4 is used instead of the expected factor of 2 because the back
stress on the source arises from dislocations piled up on both sides of the
source.
A piledup array of n dislocations can be considered for many purposes
to be a giant dislocation with Burgers vector nb. At large distances from
the array the stress due to the dislocations can be considered to be due
to a dislocation of strength nb located at the center of gravity three
quarters of the distance from the source to the head of the pileup. The
total slip produced by a pileup can be considered that due to a single
dislocation nb moving a distance 3L/4. Very high forces act on the dis
1 J. D. Eshelby, F. C. Frank, and F. R. N. Nabarro, Phil. Mag., vol. 42, p. 351,
1951; calculations for more complicated types of pileups have been given by A. K.
Head, Phil. Mag., vol. 4, pp. 295302, 1959; experimental confirmation of theory has
been obtained by Meakin and Wilsdorf, op. cit., pp. 745752.
188 Metallurgical Fundamentals [Chap. 6
locations at the head of the pileup. This force is equal to nbrg, where
T, is the average resolved shear stress on the slip plane. Koehler^ has
pointed out that large tensile stresses of the order of nr will be produced
at the head of a pileup. Stroh has made a somewhat more detailed
analysis of the stress distribution at the head of a dislocation pileup.
Using the coordinate system given in Fig. 617, he showed that the tensile
stress normal to a line OP is given by
m
 ) Ts sin d cos 2 (631)
The maximum value of a occurs at cos 9 = i^ or 6 = 70.5°. For this
situation
(632)
The shear stress acting in the plane OP is given by
T = ^Ts (jY (633)
where /S is an orientationdependent factor which is close to unity.
The number of dislocations which can be supported by an obstacle
will depend on the type of barrier, the orientation relationship between
the slip plane and the structural features at the barrier, the material, and
the temperature. Breakdown of a barrier can occur by slip on a new
plane, by climb of dislocations around the barrier, or by the generation
of high enough tensile stresses to produce a crack.
Fetch's equation that expresses the dependence of yield stress on grain
size can be developed from the concepts discussed above. Yielding is
assumed to occur when a critical shear stress Tc is produced at the head
of the pileup. This stress is assumed independent of grain size. From
Eq. (630) we get
7r(l  pWD
nTs =
4G6
It is assumed that the resolved shear stress is equal to the applied stress
minus the average internal stress required to overcome resistances to dis
location motion. If, in addition, shear stresses are converted to uniaxial
tensile stresses, for example, Tc = oe/2, the above expression becomes
7r(l  p)(ao <r,)£ ) _
8Gb
1 J. S. Koehler, Phys. Rer., vol. 85, p. 480, 1952.
2 A. N. Stroh, Proc. Roy. Soc. (London), vol. 223, pp. 404414, 1954.
Sec. 616] Dislocation Theory 189
This can be rearranged to give the desired relationship between yield
stress (To and grain diameter D.
 = " + VJ^) 5 =  + "'"' ^"''^
BIBLIOGRAPHY
Burgers, J. M., and W. G. Burgers: Dislocations in Crystal Lattices, in F. R. Eirich
(ed.), "Rheology," vol. I, Academic Press Inc., New York, 1956.
Cohen, M. (ed.): "Dislocations in Metals," American Institute of Mining and Metal
lurgical Engineers, New York, 1953.
Cottrell, A. H.: "Dislocations and Plastic Flow in Crystals," Oxford University Press,
New York, 1953.
Fisher, J. C, W. G. Johnston, R. Thomson, and T. Vreeland, Jr. (eds.): "Dislocations
and Mechanical Properties of Crystals," John Wiley & Sons, Inc., New York,
1957.
Read, W. T., Jr.: "Dislocations in Crystals," McGrawHill Book Company, Inc.,
New York, 1953.
Schoek, G.: Dislocation Theory of Plasticity of Metals, in "Advances in Applied
Mechanics," vol. IV, Academic Press, Inc., New York, 1956.
Van Bueren, H. G.: "Imperfections in Crystals," Interscience Publishers, Inc., New
York, 1960.
Chapter 7
FRACTURE
71 . Introduction
Fracture is the separation, or fragmentation, of a solid body into two
or more parts under the action of stress. The process of fracture can be
considered to be made up of two components, crack initiation and crack
propagation. Fractures can be classified into two general categories,
ductile fracture and brittle fracture. A ductile fracture is characterized
by appreciable plastic deformation prior to and during the propagation
of the crack. An appreciable amount of gross deformation is usually
present at the fracture surfaces. Brittle fracture in metals is character
ized by a rapid rate of crack propagation, with no gross deformation and
very little microdeformation. It is akin to cleavage in ionic crystals.
The tendency for brittle fracture is increased with decreasing temper
ature, increasing strain rate, and triaxial stress conditions (usually pro
duced by a notch). Brittle fracture is to be avoided at all cost, because
it occurs without warning and usually produces disastrous consequences.
This chapter will present a broad picture of the fundamentals of the
fracture of metals. Since most of the research has been concentrated
on the problem of brittle fracture, this topic will be given considerable
prominence. The engineering aspects of brittle fracture will be con
sidered in greater detail in Chap. 14. Fracture occurs in characteristic
ways, depending on the state of stress, the rate of application of stress,
and the temperature. Unless otherwise stated, it will be assumed in this
chapter that fracture is produced by a single application of a uniaxial
tensile stress. Fracture luider more complex conditions will be con
sidered in later chapters. Typical examples are fracture due to torsion
(Chap. 10), fatigue (Chap. 12), and creep (Chap. 13) and lowtemper
ature brittle fracture, temper embrittlement, or hydrogen embrittlement
(Chap. 14).
72. Types of Fracture in Metals
Metals can exhibit many different types of fracture, depending on the
material, temperature, state of stress, and rate of loading. The two
190
72]
Fracture
191
broad categories of ductile and brittle fracture have already been con
sidered. Figure 71 schematically illustrates some of the types of tensile
fractures which can occur in metals. A brittle fracture (Fig. 7la) is
characterized by separation normal to the tensile stress. Outwardly
there is no evidence of deformation, although with Xray diffraction
analysis it is possible to detect a
thin layer of deformed metal at the
fracture surface. Brittle fractures
have been observed in bcc and hep
metals, but not in fee metals unless
there are factors contributing to
grainboundary embrittlement.
Ductile fractures can take several
forms. Single crystals of hep metals
may slip on successive basal planes {a)
.■O's
I
Fig. 71. Tj^pes of fractures observed in
metals subjected to uniaxial tension,
(a) Brittle fracture of single crystals
and polycrystals; (6) shearing fracture
in ductile single crystals; (c) completely
ductile fracture in polycrystals; (d)
ductile fracture in polycrystals.
until finally the crystal separates by
shear (Fig. 716). Polycrystalline
specimens of very ductile metals, like
gold or lead, may actually be drawn
down to a point before they rupture
(Fig. 7lc). In the tensile fracture
of moderately ductile metals the
plastic deformation eventually produces a necked region (Fig. 7 Id).
Fracture begins at the center of the specimen and then extends by a
shear separation along the dashed lines in Fig. 7 Id. This results in the
familiar "cupandcone" fracture.
Fractures are classified with respect to several characteristics, such as
strain to fracture, crystallographic mode of fracture, and the appearance
of the fracture. Gensamer^ has summarized the terms commonly used
to describe fractures as follows:
Behavior described
Terms used
Crystallographic mode
Appearance of fracture
Shear
Fibrous
Ductile
Cleavage
Granular
Strain to fracture . .
Brittle
A shear fracture occurs as the result of extensive slip on the active
slip plane. This type of fracture is promoted by shear stresses. The
cleavage mode of fracture is controlled by tensile stresses acting normal
to a crystallographic cleavage plane. A fracture surface which is caused
' M. Gensamer, General Survey of the Problem of Fatigue and Fracture, in "Fatigue,
and Fracture of Metals," John Wiley & Sons, Inc., New York, 1952.
192 Metallursical Fundamentals
[Chap. 7
by shear appears at low magnification to be gray and fibrous, while a
cleavage fracture appears bright or granular, owing to reflection of light
from the flat cleavage surfaces. Fracture surfaces frequently consist of
a mixture of fibrous and granular fracture, and it is customary to report
the percentage of the surface area represented by one of these categories.
Based on metallographic examination, fractures in polycrystalline sam
ples are classified as either transgranular (the crack propagates through
the grains) or inter granular (the crack propagates along the grain bound
aries). A ductile fracture is one which exhibits a considerable degree of
deformation. The boundary between a ductile and brittle fracture is
arbitrary and depends on the situation being considered. For example,
nodular cast iron is ductile when compared with ordinary gray iron; yet
it would be considered brittle when compared with mild steel. As a
further example, a deeply notched tensile specimen will exhibit little
gross deformation; yet the fracture could occur by a shear mode.
73. Theoretical Cohesive Strength oF Metals
Metals are of great technological value, primarily because of their high
strength combined with a certain measure of plasticity. In the most
basic terms the strength is due to
the cohesive forces between atoms.
In general, high cohesive forces are
related to large elastic constants,
high melting points, and small coeffi
cients of thermal expansion. Figure
72 shows the variation of the cohe
sive force between two atoms as a
function of the separation between
these atoms. This curve is the re
sultant of the attractive and repul
sive forces between the atoms. The
interatomic spacing of the atoms in
the unstrained condition is indicated
by ao. If the crystal is subjected to a tensile load, the separation between
atoms will be increased. The repulsive force decreases more rapidly with
increased separation than the attractive force, so that a net force between
atoms balances the tensile load. As the tensile load is increased still
further, the repulsive force continues to decrease. A point is reached
where the repulsive force is negligible and the attractive force is decreas
ing because of the increased separation of the atoms. This corresponds
to the maximum in the curve, which is equal to the theoretical cohesive
strength of the material.
Separation
between
atoms, X
Fig. 72. Cohesive force as a function of
*^^he separation between atoms.
Sec. 73] Fracture 193
A good approximation to the theoretical cohesive strength can be
obtained if it is assumed that the cohesive force curve can be repre
sented by a sine curve.
o = o„,a, sm —  (71)
where omax is the theoretical cohesive strength. The work done during
fracture, per unit area, is the area under the curve.
f/o = / <T„ax sm ^ dx. = —^^ (72)
J3 A TT
The energy per unit area required to produce a new surface is y. If it is
assumed that all the work involved in fracture goes into creating two new
surfaces, Eq. (72) can be written
27r'Y
or (T^a. = — (73)
Since Hooke's law holds for the initial part of the curve, the stress can
be written as
<T = — (74)
do
In order to eliminate X from Eq. (73), take the first derivative of Eq.
(71).
da _ 2tt 2wx
dx ~ '"'"="' y ^°^ IT
Since cos (27rx/X) is approximately unity for the small values of x which
are involved, the above expression can be written as
Also, Eq. (74) can be differentiated to give
dx ao
Equating (75) and (76) and substituting into Eq. (73) produces the
final expression for the theoretical cohesive strength of a crystal.
O"
(77)
194 Metallurgical Fundamentals
[Ch
ap.
The substitution of reasonable values for the quantities involved in
the above expression (see Prob. 71) results in the prediction of a cohesive
strength of the order of 2 X 10^ psi. This is 10 to 1,000 times greater
than the observed fracture strengths of metals. Only the fracture
strength of dislocationfree metal whiskers approaches the theoretical
cohesive strength.
74. Griffith Theory of Brittle Fracture
The first explanation of the discrepancy between the observed fracture
strength of crystals and the theoretical cohesive strength was proposed
by Griffith.^ Griffith's theory in its original form is applicable only to
a perfectly brittle material such as glass. However, while it cannot be
applied directly to metals, Griffith's ideas have had great influence on
the thinking about the fracture of metals.
Griffith proposed that a brittle material contains a population of fine
cracks which produce a stress concentration of sufficient magnitude so
that the theoretical cohesive strength is reached
in localized regions at a nominal stress which is
well below the theoretical value. When one of
the cracks spreads into a brittle fracture, it pro
duces an increase in the surface area of the sides
of the crack. This requires energy to overcome
the cohesive force of the atoms, or, expressed in
another way, it requires an increase in surface en
ergy. The source of the increased surface energy
is the elastic strain energy which is released as the
crack spreads. Griffith established the following
criterion for the propagation of a crack : A crack
will propagate when the decrease in elastic strain
energy is at least equal to the energy required to
create the new crack surface. This criterion can be
used to determine the magnitude of the tensile
stress w^hich will just cause a crack of a certain
size to propagate as a brittle fracture.
Consider the crack model shown in Fig. 73. The thickness of the
plate is negligible, and so the problem can be treated as one in plane stress.
The cracks are assumed to have an elliptical cross section. For a crack
at the interior the length is 2c, while for an edge crack it is c. The effect
of both types of crack on the fracture behavior is the same. The stress
1 A. A. Griffith, Phil. Trans. Roy. Soc. London, vol. 221A, pp. 163198 1920; First
Intn. Congr. Appl. Mech., Delft, 1924, p. 55.
Fig. 73. Griffith crack
model.
Sec. 74] Fracture 195
distribution for an elliptical crack was determined by Inglis.^ A decrease
in strain energy results from the formation of a crack. The elastic strain
energy per unit of plate thickness is equal to
U. =  "^ (78,
where a is the tensile stress acting normal to the crack of length 2c. The
surface energy due to the presence of the crack is
Us = 4c7 (79)
The total change in potential energy resulting from the creation of the
crack is
AU = Us+ Ue (710)
According to Griffith's criterion, the crack will propagate under a con
stant applied stress a if an incremental increase in crack length produces
no change in the total energy of the system; i.e., the increased surface
energy is compensated by a decrease in elastic strain energy.
dAU _ = —
dc dc
(^^^  ^^)
Equation (711) gives the stress required to propagate a crack in a brittle
material as a function of the size of the microcrack. Note that this
equation indicates that the fracture stress is inversely proportional to
the square root of the crack length. Thus, increasing the crack length
by a factor of 4 reduces the fracture stress by onehalf.
For a plate which is thick compared with the length of the crack (plane
strain) the Griffith equation is given by
2Ey
(1  j')Vc
(712)
Analysis of the threedimensional case, where the crack is a very flat
oblate spheroid, results only in a modification to the constant in Griffith's
equation. Therefore, the simplification of considering only the two
dimensional case introduces no large error.
An alternative way of rationalizing the low fracture strength of solids
1 C. E. Inglis, Trans. Inst. Naval Architects, vol. 55, pt. I, pp. 219230, 1913.
2 R. A. Sack, Proc. Phys. Soc. (London), vol. 58, p. 729, 1946.
196 Metallurgical Fundamentals [Chap. 7
with the high theoretical cohesive strength was proposed by Orowan.^
Inghs showed that the stress at the end of an ellipsoidal crack of length
2c, with a radius of curvature p at the end of the crack, is given by
(713)
where o is the nominal stress when no crack is present. The sharpest
radius of curvature at the end of the crack should be of the order of the
interatomic spacing, p = ao. Making this substitution in Eq. (713) and
combining it with Eq. (77) results in an expression for the critical stress
to cause brittle fracture which is similar to Griffith's equation.
m'
(714)
Within the accuracy of the estimate, this equation predicts the same
stress needed to propagate a crack through a brittle solid as the Griffith
equation.
Griffith's theory satisfactorily predicts the fracture strength of a com
pletely brittle material such as glass. ^ In glass, reasonable values of
crack length of about 1 n are calculated from Eq. (711). For zinc, the
theory predicts a crack length of several millimeters. This average
crack length could easily be greater than the thickness of the specimen,
and therefore the theory cannot apply.
Early experiments on the fracture of glass fibers showed that strengths
close to the theoretical fracture strength could be obtained with fibers
freshly drawn from the melt. The highest fracture strengths were found
with the smallestdiameter fibers, since on the average these fibers would
have the shortest microcracks. However, other factors besides diam
eter, such as method of preparation, temperature of the melt, and
amount and rate of drawing from the melt, can affect strength. Recent
results^ indicate that there is no dependence of strength on diameter
when differentsize glass fibers are prepared under nearly identical con
ditions. Experiments on metal whiskers* have also demonstrated frac
ture strengths close to the theoretical value. The strength of a metal
whisker varies inversely with its diameter. This is the type of size
dependence that would be expected if the strength were controlled by
the number of surface defects. On the other hand, if the whisker con
tains a certain number of dislocation sources, the length of the most
1 E. Orowan, Welding J., vol. 34, pp. 157s160s, 1955.
2 O. L. Anderson, The Griffith Criterion for Glass Fracture, in "Fracture," pp. 331
353, John Wiley & Sons, Inc., New York, 1959.
3 F. Otto, /. Am. Ceramic Soc, vol. 38, p. 123, 1955.
< S. S. Brenner, J. Appl. Phys., vol. 27, p. 1484, 1956.
Sec. 75] Fracture 197
extended source will vary directly with the diameter and the strength
will again be inversely related to the whisker diameter. Thus, it is not
possible from the size dependence of strength to establish whether the
high strength of whiskers is due to a freedom from surface defects or
dislocations.
The strength of glass fibers is extremely sensitive to surface defects.
If the surface of a freshly drawn fiber is touched with a hard object, the
strength will instantly decrease to a low value. Even the strength of
a fiber which has not been handled will, under the influence of atmos
pheric attack, decrease to a low value within a few hours of being drawn
from the melt.
Joffe^ showed that the fracture strength of NaCl crystals could be
greatly increased when the test was carried out under water. This
Joffe effect has been attributed to the healing of surface cracks by the
solution of the salt crystal in the water. The fracture behavior of other
ionic crystals has been shown to depend on the environment in contact
with the surface. However, the Joffe effect in these crystals cannot
always be explained simply by surface dissolution.
75. ModiFications of the Griffith Theory
Metals which fracture in a brittle manner show evidence of a thin layer
of plastically deformed metal when the fracture surface is examined by
Xray diffraction. Other indications that brittle fracture in metals is
always preceded by a small amount of plastic deformation, on a micro
scopic scale, are given in Sec. 77. Therefore, it appears that Griffith's
theory, in its original form, should not be expected to apply to the brittle
fracture of metals.
Orowan' suggested that the Griffith equation could be made more
compatible with brittle fracture in metals by the inclusion of a term p
expressing the plastic work required to extend the crack wall.
2E(y + p)
ire
m'
(715)
The surfaceenergy term can be neglected, since estimates of the plastic
work term are about 10^ to 10^ ergs/cm^, compared with values of y of
1 A. F. Joffe, "The Physics of Crystals," McGrawHill Book Company, Inc., New
York, 1928.
2E. P. Klier, Trans. ASM, vol. 43, pp. 935957, 1951; L. C. Chang, /. of Mech.
andPhys. Solids, vol. 3, pp. 212217, 1955; D. K. Felbeck and E. Orowan, Welding J.,
vol. 34, pp. 570s575s, 1955.
' E. Orowan, in "Fatigue and Fracture of Metals," symposium at Massachusetts
Institute of Technology, John Wiley & Sons, Inc., New York, 1950.
198 Metallurgical Fundamentals [Chap. 7
about 1,000 to 2,000 ergs/cm. There is some experimental evidence
that p decreases with decreasing temperature.
An extension of the Griffith theory into the area of fracture mechanics
has been made by Irwin. ^ The objective is to find a reHable design cri
terion for predicting the stress at which rapidly propagating fractures
will occur. This is essentially a macroscopic theory that is concerned
with cracks that are tenths of an inch in length or greater. The quantity
of interest is the crackextension force, also called the strainenergy release
rate. The crackextension force 9, measured in units of in.lb/in.^, is the
quantity of stored elastic strain energy released from a cracking specimen
as the result of the extension of an advancing crack by a unit area. When
this quantity reaches a critical value £c, the crack will propagate rapidly.
9c is the fracture toughness. It represents the fraction of the total work
expended on the system which is irreversibly absorbed in local plastic
flow and cleavage to create a unit area of fracture surface. 9c appears
to be a basic material property which is essentially independent of size
effects. It does depend on composition, microstructure, temperature,
and rate of loading. Values of 9c for steel vary from about 100 to
600 in. lb/in. 2, depending on temperature and composition.
To measure 9c, it is necessary to have a reliable mathematical expres
sion for 9 in terms of the crack dimensions, the geometry of the speci
men, the elastic constants, and the nominal applied stress.^ The speci
men is then loaded until a stress is reached at which the crack which was
initially present in the specimen propagates rapidly. The calculated
value of 9 for this condition is equal to 9c For a crack of length 2c in an
infinitely wide plate the relationship between the stress and 9 is given by
'  &r
Comparison of Eq. (716) with the modified Griffith equation (715)
shows that 9 is analogous to Orowan's plasticwork factor p. In the
original Griffith theory, a crack was assumed to propagate rapidly when
9 = 27. However, in Irwin's modification of this theory 9 is taken as
an experimentally determined parameter For a finite plate of width L
with a central crack of length 2c or two edge cracks of length c, the crack
extension force for tensile loading is given by
= ^ (1  .^) tan (^) (717)
E
^ G. R. Irwin, Naval Research Lab. Rept. 4763, May, 1956, available from Office of
Technical Services, PB 121224; G. R. Irwin, J. A. Kies, and H. L. Smith, Proc. ASTM,
vol. 58, pp. 640660, 1958.
2 Detailed procedures for measuring 9c in tension have been presented in the ASTM
Bulletin, January and February, 1960. Methods using a notchedbend test and a high
speed rotating disk have been given by D. H. Winne and B. M. Wundt, Trans.
ASME, vol. 80, p. 1643, 1958.
Sec. 76]
Fracture
199
76. Fracture of Sinsle Crystals
The brittle fracture of single crystals is considered to be related to the
resolved normal stress on the cleavage plane. Sohncke's law states that
fracture occurs when the resolved normal stress reaches a critical value.
Considering the situation used to develop the resolved shear stress for
slip (Fig. 418), the component of the tensile force which acts normal to
the cleavage plane is P cos </>, where is the angle between the tensile axis
and the normal to the plane. The area of the cleavage plane is A /(cos <^).
Therefore, the critical normal stress for brittle fracture is
P cos </) P
A/(cos 4>) A
= r cos"
(718)
The cleavage planes for certain metals and values of the critical normal
stress are given in Table 71.
Table 71
Critical Normal Stress for Cleavage of Single Crystals t
Metal
Crystal
lattice
Cleavage plane
Temper
ature,
°C
Critical
normal
stress,
kg/mm^
Iron
boo
hep
hep
hep
hep
Hexagonal
Rhombohedral
Rhombohedral
(100)
(0001)
(0001)
(0001)
(0001), (1011)
(1012), (1010)
(1010)
(111)
(111)
100
185
185
185
185
20
20
20
26
Zinc (0.03% Cd)
Zinc (0.13% Cd)
Zinc (0.53 % Cd)
Magnesium
27.5
0.19
0.30
1.20
Tellurium
0.43
Bismuth
32
Antimony
66
t Data from C. S. Barrett, "Structure of Metals," 2d ed., McGrawHill Book Com
pany, Inc., New York, 1952; N. J. Fetch, The Fracture of Metals, in "Progress in
Metal Physics," vol. 5, Fergamon Press, Ltd., London, 1954.
Although Sohncke's law has been accepted for over 25 years, it is not
based on very extensive experimental evidence. Doubt was cast on its
reliability by fracture studies^ on zinc single crystals at —77 and — iy6°C.
The resolved normal cleavage stress was found to vary by over a factor
of 10 for a large difference in orientation of the crystals. This vari
ation from the normalstress law may be due to plastic strain prior to
fracture, although it is doubtful that this could account for the observed
discrepancy.
1 A. Deruyttere and G. B. Greenough, /. Inst. Metals, vol. 84, pp. 337345, 195556.
200 Metallurgical Fundamentals [Chap. 7
Several modes of ductile fracture in single crystals are shown in Fig.
71. Under certain conditions hep metals tested at room temperature
or above will shear only on a restricted number of basal planes. Frac
ture will then occur by "shearing off" (Fig. 716). More usually, slip
will occur on systems other than the basal plane, so that the crystal
necks down and draws down almost to a point before rupture occurs.
The usual mode of fracture in fee crystals is the formation of a necked
region due to multiple slip, followed by slip on one set of planes until
fracture occurs. The crystal can draw down to a chisel edge or a point
(if multiple slip continues to fracture). The best stress criterion for
ductile fracture in fee metals appears to be the resolved shear stress on
the fracture plane (which is usually the slip plane).
The mode of fracture in bcc iron crystals is strongly dependent on
temperature, purity, heat treatment, and crystal orientation. ^ Crystals
located near the [001] corner of the stereographic triangle show no measur
able ductility when tested in tension at — 196°C, while crystals closer to
[111] and [Oil] orientations may rupture by drawing down to a chisel edge
when tested at the same temperature. An interesting point is that the
change from brittle to ductile fracture is very sharp, occurring over a
change in orientation of only about 2°.
77. Metallosraphic Aspects of Brittle Fracture
Because of the prominence of the Griffith theory, it has been natural
for metallurgists to use their microscopes in a search for Griffith cracks in
metals. However, based on observations up to the magnifications avail
able with the electron microscope, there is no reliable evidence that
Griffith cracks exist in metals in the unstressed condition. There is,
however, a growing amount of experimental evidence to show that
microcracks can be produced by plastic deformation.
Metallographic evidence of the formation of microcracks at nonmetallic
inclusions in steel as a result of plastic deformation has existed for a num
ber of years. These microcracks do not necessarily produce brittle frac
ture. However, they do contribute to the observed anisotropy in the
ductilefracture strength. The fact that vacuummelted steel, which is
very low in inclusions, shows a reduction in the fracture anisotropy sup
ports the idea of microcracks being formed at secondphase particles.
An excellent correlation between plastic deformation, microcracks, and
brittle fracture was made by Low.^ He showed that for mild steel of a
1 N. P. Allen, B. E. Hopkins, and J. E. McLennan, Proc. Roy. Soc. (London), vol.
234A, p. 221, 1956.
2 J. R. Low, I.U.T.A.M. Madrid Colloqium, "Deformation and Flow of Solids,"
p. 60, Springer Verlag OHG, Berlin, 1956.
Sec. 77] Fracture 201
given grain size tested at — 196°C brittle fracture occurs in tension at
the same value of stress that is required to produce yielding in com
pression. Microcracks only one or two grains long were observed. More
detailed studies of the conditions for microcrack formation have been
made^ with tensile tests on mild steel at carefully controlled subzero tem
peratures. Figure 74 illustrates a typical microcrack found in a speci
men before it fractured.
The correlation between the temperature dependence of yield stress,
^^^0^
/'^
Fig. 74. Microcracks produced in iron by tensile deformation at — 140°C. 250 X
{Courtesy G. T. Hahn.)
fracture stress, and ductility and microcrack formation is shown in Fig.
75. In region A, in the neighborhood of room temperature, a tensile
specimen fails with a ductile cupandcone fracture. The reduction of
area at fracture is of the order of 50 to 60 per cent. In region B the
fracture is still ductile, but the outer rim of the fracture contains cleavage
facets. A transition from ductile to brittle fracture occurs at the duc
tility transition temperature Td. The existence of a transition temper
ature is indicated by the drop in the reduction of area at the fracture
to practically a zero value. Accompanying this is a large decrease in
the fracture stress. The percentage of grains containing microcracks
increases rapidly in region C just below Td. However, microcracks are
found above Td. Therefore, the ductility transition occurs when the
conditions are suitable for the growth of microcracks into propagating
fractures. The initiation of microcracks is not a sufficient criterion for
• G. T. Hahn, W. S. Owen, B. L. Averbach, and M. Cohen, Welding J., vol. 38,
pp. 367s376s, 1959.
202 Metallurgical Fundamentals
[Chap. 7
brittle fracture. Microcracks occur only in regions which have under
gone discontinuous yielding as a result of being loaded through the upper
yield point. As the temperature drops in region C, eventually the frac
ture stress drops to a value equal to the lower yield stress. In region D
the lower yield stress and fracture stress are practically identical. Frac
ture occurs at a value equal to the lower yield stress after the material
CJ» o
t o
Lower yield ^
stress 
Reduction
in area
tl^icrocraclfs
100
Temoeroture, °C
Fig. 75. Temperature dependence of fracture stress, yield stress, and microcrack fre
quency for mild steel. {After G. T. Hahn, W. S. Owen, B. L. Averbach, and M. Cohen,
Welding J., vol. 38, p. 372s, 1959.)
has undergone some discontinuous yielding. The fracture stress increases
because the yield stress is increasing with decreasing temperature. In
region E cleavage fracture occurs abruptly before there is time for dis
continuous yielding. Presumably fracture occurs at the first spot to
undergo discontinuous yielding. Finally, at very low temperatures in
region F fracture is initiated by mechanical twins. Mechanical twins are
observed at temperatures as high as Td, but it is only in region F that
they appear to be the source of initiation of fracture.
Detailed experiments such as these demonstrate that the cracks respon
sible for brittlecleavagetype fracture are not initiallj' present in the
material but are produced by the deformation process. The fact that
at appropriate temperatures appreciable numbers of microcracks are
Sec. 77]
Fractu
re
203
present shows that the conditions for the initiation of a crack are not
necessarily the same as conditions for the propagation of a crack. The
process of cleavage fracture should be considered to be made up of
three steps, (1) plastic deformation, (2) crack initiation, and (3) crack
propagation.
Most brittle fractures occur in a transgranular manner. However, if
the grain boundaries contain a film of brittle constituent, as in sensitized
7^^
Fig. 76. Cleavage steps and river pattern on a cleavage surface.
austenitic stainless steel or molybdenum alloys containing oxygen, nitro
gen, or carbon, the fracture will occur in an intergranular manner. Inter
granular failure can also occur without the presence of a microscopically
visible precipitate at the grain boundaries. Apparently, segregation at
the grain boundaries can lower the surface energy sufficiently to cause
intergranular failure. The embrittlement produced by the addition of
antimony to copper and oxygen to iron and the temper embrittlement
of alloy steels are good examples.
Sometimes a considerable amount of information can be obtained by
examining the surfaces of the fracture at fairly high magnifications. This
type of examination is known as fractography.^ At high magnification,
transgranularcleavage surfaces usually contain a large number of cleav
age steps and a "river pattern" of branching cracks (Fig. 76). These
are indications of the absorption of energy by local deformation. The
' C. A. Zappfe and C. O. Worden, Trans. ASM, vol. 42, pp. 577603, 1950.
204 Metallurgical Fundamentals [Chap. 7
surfaces of intergranular brittle fractures are mvich smoother, with a
general absence of cleavage steps. From the appearance of the fracture
surface, the energy absorbed in an intergranular fracture is much lower
than for transgranular cleavage.
78. Dislocation Theories of Fracture
The idea that the high stresses produced at the head of a dislocation
pileup could produce fracture was first advanced by Zener.^ The shear
stress acting on the slip plane squeezes the dislocations together. At
some critical value of stress the dislocations at the head of the pileup
are pushed so close together that they coalesce into an embryonic crack
or cavity dislocation. After analyzing the stresses at a dislocation pileup
and making use of the Griffith criterion, Stroh^ has proposed that a cleav
age crack can form when n dislocations piled up under the action of a
resolved shear stress r^ satisfy the condition
nbTs = 127 (719)
where b is the Burgers vector and y is the surface energy. The length
of the slip plane that the pileup will occupy is given by
nbG
7r(l — v)ts
Eliminating n from these equations gives
(720)
rs^L = ^^ (721)
7r(l — V)
When a specimen of grain size D is tested in tension, Ts = (t/2 and
L = D/2. The fracture stress in tension can be expressed in terms of
grain size by
QGy
0/
7r(l  v)
' D'' = KD^'^ (722)
However, Petch^ has found that experimental data for iron and steel
agree best with an equation of the type
as = a + KD^'^ (723)
^ C. Zener, The Micromechanism of Fracture, in "Fracturing of Metals," American
Society for Metals, Metals Park, Ohio, 1948.
2 A. N. Stroh, Proc. Roy. Soc. (London), vol. 223 A, p. 404, 1954; Phil. Mag., vol. 46,
p. 968, 1955.
3 N. J. Fetch, /. Iron Steel Inst. {London), vol. 174, p. 25, 1953.
Sec. 78] Fracture 205
This equation is very similar to the equation expressing the grainsize
dependence of the yield strength.
(TO = cr, + KyD^'^ (724)
This similarity is to be expected in view of the fact that yielding and
brittle fracture are closely related. In both equations ai is the frictional
stress resisting the motion of an unlocked dislocation. This term
increases with decreasing temperature of testing. The constant K in
the fracture equation is given approximately by Eq. (722). The con
stant Ky in the yieldstrength equation is a measure of the localized stress
needed to unlock dislocations held up at a grain boundary so that yield
ing can be transmitted to the next grain by the propagation of a Liiders
band. This quantity is important in current theories of fracture.
The fact that brittle fracture can occur in single crystals suggests that
the role of grain boundaries as barriers for dislocation pileup may be
overemphasized in current theories. Also, it is questionable that the
necessary stress concentration can be produced at the head of a pileup
before slip occurs in the neighboring grains to relieve the high localized
stresses. It is possible that deformation twins may act as barriers for
dislocation pileup. For example, the strong orientation dependence of
the brittle fracture of iron single crystals can be explained' on this basis.
While there is experimental evidence to indicate that twin intersections
may initiate brittle fracture, there is also evidence to show that brittle
fracture can be produced in the absence of mechanical twins. Another
mechanism by which cracks can form is the glide of dislocations on inter
secting slip planes according to the hypothesis of CottrelP (see Sec. 66
and Fig. 69). This mechanism is energetically favorable for a bcc and
hep metal, but not for an fee lattice, in agreement with the fact that
fee metals do not undergo brittle fracture.
A consideration of the known facts of fracture has led Cottrell and
Fetch independently to conclude that the growth of a microcrack into a
selfpropagating fracture is a more difficult step than the nucleation of
microcracks from glide dislocations. Support for this viewpoint is found
in the fact that many nonpropagating microcracks are observed. More
over, crack nucleation by dislocation coalescence should depend only on
the shear stress, not the hydrostatic component of stress. But there is
ample experimental evidence that fracture is strongly influenced by the
hydrostatic component of stress (see Sec. 716). If the propagation of
microcracks, according to a Griffithtype criterion, is the controlling step
in fracture, the stress normal to the crack would be an important factor.
1 H. K. Birnbaum, Acta Met., vol. 7, pp. 516517, 1959.
2 D. Hull, Acta Met., vol. 8, pp. 1118, 1960.
3 A. H. Cottrell, Trans. Met. Soc. AIME, vol. 212, pp. 192203, 1958.
206 Metallurgical Fundamentals [Chap. 7
This should lead to a strong dependence of fracture on the hydrostatic
component of stress.
Utilizing the Griffith criterion, Cottrell^ has shown that the stress
required to propagate a microcrack is given by
.^ (725)
where n is the number of dislocations of Burgers vector h that coalesce
into the crack and y is the surface energy of the crack. To evaluate nh,
assume that a slip plane of length L is acted on by an applied shear stress
r ~ a/2. The effective shear stress on the slip plane is given by r — n,
where n is the frictional resistance. The shear displacement at the center
of the length L is given by (r — Ti)L/G, and this is approximately equal to
nh. If L is taken as about onehalf the average grain diameter D,
Equation (724) can be written in terms of shear stress as
TO = r. + k,D'^^ (727)
Writing Eq. (725) as nhro = y and substituting for nh and to from the
above equations results in
inD'^' + k,)k, = (?7/3 (728)
or the equivalent relationship
TokyDy^ = Gyl3 (729)
In the above equations /3 is a term which expresses the ratio of the maxi
mum shear stress to the maximum normal stress. For torsion /3 = 1,
for tension jS = 3^^, and for the plastically constrained region at the root
of a notch j8 « 3^.
When the glide dislocations coalesce into a crack or a cavity disloca
tion, the frictional resistance to glide equals zero. Therefore, by making
substitutions from the above equations into Eq. (725) one arrives at an
expression for the stress required to propagate a microcrack of length D.
.  2 (^^y (730)
Equations (728) and (729) express the limiting conditions for the for
mation of propagating crack from a pileup of glide dislocations. If con
ditions are such that the lefthand side of the equation is less than the
» Ihid.
78]
Fract
ure
207
righthand side, a crack can form but it cannot grow beyond a certain
length. This is the case of nonpropagating microcracks. When the left
hand side of the equations is greater than the right side, a propagating
brittle fracture can be produced at a shear stress equal to the yield stress.
Therefore, these equations predict a ductiletobrittle transition, such
as was shown in Fig. 75 for tension tests on mild steel at decreasing
temperature.
The equations describing the ductiletobrittle transition are expressed
in terms of the following metallurgical or mechanical factors: grain size,
state of stress, surface energy, yield stress, friction stress, and k,,. The
parameter ky is very important, since it determines the number of dis
locations that are released into a pileup when a source is unlocked.
Table 72 gives some typical values of ky obtained from measurements of
Table 72
Values of ky/Gf
Material
Temperature,
°K
ky/G,
cm '/^
Iron
Molybdenum
Columbium
Tantalum
300
300
200
200
0.4 X 10"
0.55 X 10"
0.1 X 10"
0.1 X 10"
t A. H. Cottrell, Trans. Met. Soc. AIME, vol. 212, p. 194, 1958.
fracture stress vs. grain size. Large values of ky indicate brittle behavior,
which agrees with the observations that pure columbium and tantalum
are less prone to brittle fracture than the other bcc metals, iron and
molybdenum. Equation (728) shows that at a constant temperature
there is a certain grain size above which the metal will be brittle and
below which it will be ductile. This is shown in Fig. 77, where above a
certain grain size there is measurable ductility at fracture. The fric
tional resistance n increases with decreasing temperature. However,
since this term enters into Eq. (728) as the product of Z)'^, it can be seen
that a finegrained metal can withstand higher values of r, (lower temper
atures) before becoming brittle. Many of the effects of the composition
of steel on the ductiletobrittle transition are due to changes in grain size
or in ky or r,. For example, manganese decreases the grain size and
reduces ky, while silicon produces larger grain size and increases r,.
The yield stress increases with decreasing temperature, and in agree
ment with Eq. (729) this increases the tendency for brittle fractvire. If
conditions are such that microcracks cannot propagate at the yield point,
it is necessary to increase the stress by At in order to produce fracture.
208 Metallursical Fundamentals
From Eq. (729), the necessary value of shear stress is
T/ = To + Ar = ^ DH
[Chap. 7
(731)
This predicts that fracture stress is a linear function of Z)~5^^, which
extrapolates to zero at D~5^^ = 0. Figure 77 shows that this relation
ship is satisfied. In the region of grain size for which cracks propagate
as completely brittle fractures, the fracture stress equals the yield stress.
200
3
Approximate ASTM, G.S. No.
13 5 6
I i r
X Fracture stress
o Yie/d stress
o Strain to fracture
0.6 ^
0.4 o
0.2 I
^
1 2 ^ _i "^ 1 5 6
(Grain diameter)" ^2 ^ rnm~ ^2
Fig. 77. Effect of grain size on the yield and fracture stresses for a lowcarbon steel
tested in tension at — 196°C. (/. R. Low, in "Relation of Properties to Microstructure,"
American Society for Metals, Metals Park, Ohio, 1954.)
This branch of the curve extrapolates to the fracture stress for a single
crystal.
High values of surface energy tend to promote ductile fracture.
Unfortunately, this is not a factor which is readily increased, although
various environmental and metallurgical conditions may lower the sur
face energy. The embrittlement of steel due to hydrogen has been
attributed to this factor. Intergranular fracture due to an embrittling
film may also be explained in this way.
It is well known that the presence of a notch greatly increases the
tendency for brittle fracture. The complicated effects of a notch will be
considered more fully in Sec. 712. The effect of a notch in decreasing
the ratio of shear stress to tensile stress is covered in Cottrell's equations
by the constant /3. Strain rate or rate of loading does not enter explicitly
into Cottrell's equations. However, for a notch to produce the plastic
79]
Fracture
209
constraint that results in a value of j8 c^ 3^', it is necessary for the material
to yield locally. At high rates of strain, such as occur in a notched
impact test, yielding will have to occur more rapidly. As is indicated by
Eq. (732) in the next section, this can occur at the same value of ro if the
temperature is increased. Therefore, increasing the strain rate raises the
transition temperature.
79. Delayed Yielding
A phenomenon which is important to brittle fracture is delayed yielding.
When certain metals, notably mild steel, are rapidly loaded to a constant
10
10'
10
10"2 10"' 1 10
Delay time, sec
10^
10^
10"
Fig. 78. Delay time for initiation of yielding in mild steel as a function of stress.
(D. S. Clark, Trans. ASM, vol. 46, p. 49, 1954.)
stress above the yield stress, it is found that a certain delay time is required
before plastic yielding occurs.^ Figure 78 shows that the delay time
increases with decreasing temperature at a constant stress. For a con
stant temperature, the delay time increases with decreasing stress. The
lower limiting stress shown by the horizontal portion of the curves corre
sponds to the upper yield point for tests carried out at slow speeds.
The temperature dependence of the delay time may be expressed by
an exponential relationship
t = to exp
kT
(732)
1 D. S. Clark, Trans. ASM, vol. 46, p. 34, 1954.
210 Metallurgical Fundamentals [Chap. 7
where t = delay time
^0 = a constant, approximately 10~^^ sec
A; = Boltzmann's constant
Q{a/ao) = stressdependent activation energy
Cottrell' has estimated that Qia/ao), in electron volts, is given approxi
mately by 0.9(1 — o"/a■o)^ where a is the applied stress and oq is the yield
stress.
The fact that brittle fracture occurs when plastic deformation fails to
keep the stress below a critical value indicates that there should be a
connection between delayed yielding and brittle fracture. The delay
time is quite temperaturedependent, and so is brittle fracture. In the
temperature region where brittle fracture is caused by an avalanche of
dislocations breaking away from a barrier and running together to form
a crack, delayed yielding probably plays the important role of localizing
the slip by preventing nearby dislocation sources from operating. At
temperatures where the metal fractures in a ductile manner the delay
time is so short that slip occurs around the pileups and the high localized
stresses are dissipated by plastic deformation. This agreement is sup
ported by the fact that metals which have a ductiletobrittle fracture
transition also have a delayed yield phenomenon.
710. Velocity of Crack Propagation
Brittle fracture is not possible unless the cracks which are nucleated
can propagate at a high velocity throughout the metal. Mott^ has made
an analysis of the velocity of a crack in an ideal elastic, isotropic medium.
The elastic energy that is released by the movement of the crack is the
driving force. This must be balanced by the surface energy of the new
surface that is created and the kinetic energy associated with the rapid
sidewise displacement of material on each side of the crack. The crack
velocity v is given by
V = Bvo ( 1  — ) (733)
jrhere B is a constant and Vo = {E/p)'' is the velocity of sound in the
material. The term cg is the length of a Griffith crack, as evaluated by
Eq. (711), and c is the actual crack length. When c is large compared
with Cg, Eq. (733) approaches the limiting value Bvq. The constant has
been evaluated^ for the planestress condition and found tohe B c^ 0.38.
1 A. H. Cottrell, Proc. Conf. on Properties Materials at High Rates of Strain, Institu
tion of Mechanical Engineers, London, 1957.
2 N. F. Mott, Engineering, vol. 165, p. 16, 1948.
» D. K. Roberts and A. A. Wells, Engineering, vol. 178, p. 820, 1954.
Sec. 711]
Fracture
211
Table 73 shows that experimental values for the crack velocity in brittle
materials agree quite well with the theoretical prediction that the limiting
crack velocity is given by
V = 0.38^0 = 0.38 (  ) ^" (734)
Kf)
Table 73
Velocity of Propagation of Brittle Fracture
Material
Observed
velocity,
ft /sec
v/vo
Reference
Steel
Fused quartz
Lithium fluoride
6,000
7,200
6,500
0.36
0.42
0.31
t
§
t T. S. Robertson, J. Iron Steel Inst. (London), vol. 175, p. 361, 1953.
t H. Schardin and W. Struth, Glastech. Ber., vol. 16, p. 219, 1958.
§ J. J. Oilman, C. Knudsen, and W. P. Walsh, J. Appl. Phys., vol. 29, p. 601, 1958.
711. Ductile Fracture
Ductile fracture has been studied much less extensively than brittle
fracture, probably because it is a much less serious problem. Up to this
point ductile fracture has been defined rather ambiguously as fracture
occurring with appreciable gross plastic deformation. Another impor
tant characteristic of ductile fracture, which should be apparent from
previous considerations of brittle fracture, is that it occurs by a slow
tearing of the metal with the expenditure of considerable energy. Many
varieties of ductile fractures can occur during the processing of metals
and their use in different types of service. For simplification, the dis
cussion in this section will be limited to ductile fracture of metals pro
duced in uniaxial tension. Other aspects of tensile fracture are con
sidered in Chap. 9. Ductile fracture in tension is usually preceded by
a localized reduction in diameter called necking. Very ductile metals
may actually draw down to a line or a point before separation. This
kind of failure is usually called rupture.
The stages in the development of a ductile "cupandcone" fracture
are illustrated in Fig. 79. Necking begins at the point of plastic insta
bility where the increase in strength due to strain hardening fails to
compensate for the decrease in crosssectional area (Fig. 79a). This
occurs at maximum load or at a true strain equal to the strainhardening
coefficient (see Sec. 93). The formation of a neck introduces a triaxial
state of stress in the region. A hydrostatic component of tension acts
212 Metallurgical Fundamentals
[Chap. 7
along the axis of the specimen at the center of the necked region. Many
fine cavities form in this region (Fig. 796), and under continued strain
ing these grow and coalesce into a central crack (Fig. 79c). This crack
grows in a direction perpendicular to the axis of the specimen until it
Shear
Fibrous
(^1 ie)
Fig. 79. Stages in the formation of a cupandcone fracture.
approaches the surface of the specimen. It then propagates along local
ized shear planes at roughly 45° to the axis to form the "cone" part of
the fracture (Fig. 79d).
When the central "cup" region of the fracture is viewed from above,
it has a very fibrous appearance, much as if the individual elements of
the specimen were split into longitudinal fibers and were then drawn
down to a point before rupture. When the fracture is sectioned longi
tudinally, the central crack has a zigzag contour, such as would be pro
duced by tearing between a number of holes. The outer cone of the
fracture is a region of highly localized shear. Extensive localized defor
mation occurs by the sliding of grains over one another, and because the
Sec. 71 2] Fracture 21 3
shear fracture propagates rapidly compared with the fibrous fracture,
there is appreciable localized heating.
Fetch ^ has shown that the fracture stress (corrected for necking) for
the ductile fracture of iron has the same dependence on grain size as is
found for brittle fracture. This suggests that the voids are nucleated
by dislocation pileups at grain boundaries. However, it is extremely
unlikely that dislocation pileups large enough to produce cavity dis
locations can be produced in ductile fee metals like aluminum and copper.
Instead, voids in these metals appear to be nucleated at foreign particles
such as oxide particles, impurity phases, or secondphase particles.
Under tensile strain either the metal separates from the inclusion, or
the inclusion itself fractures. ^ Even in metals for which no crack
nucleating secondphase particles can be observed, it appears that
fracturenucleating elements are present before deformation. This is
borne out by the fact that the fracture stress and reduction in area can
be appreciably lower when tested in a direction perpendicular to the
original rolling or extrusion direction than when tested in the direction
of working, even though all microstructural evidence of working has been
removed by heat treatment and there is no strong crystallographic
texture. It must be assumed that working elongates these "sites" so
that they open into voids more readily when the tensile stress is applied
perpendicular to their length.
71 2. Notch Effect in Fracture
The changes produced by the introduction of a notch have important
implications for the fracture of metals. The presence of a notch will
very appreciably increase the temperature at which a steel changes from
ductile to brittle fracture. The introduction of a notch results in a stress
concentration at the root of the notch. Figure 710 shows the non
uniform distribution of the longitudinal tensile stress in a notched tensile
bar. When yielding occurs at the root of the notch, the stress concen
tration is reduced. However, transverse and radial stresses are set up
in the vicinity of the notch (Fig. 710). The radial stress <tr is zero at
the free surface at the root of the notch, but it rises to a high value in
the interior of the specimen and then drops off again. The transverse
stress (tt acts in the circumferential direction of a cylindrical specimen.
This stress drops from a high value at the notch root to a lower value
at the specimen axis.
The occurrence of this state of stress can be explained by the con
straints to plastic flow which a notch sets up. For an equilibrium of
1 N. J. Fetch, Phil. Mag., ser. 8, vol. 1, p. 186, 1956.
2 K. E. Puttick, Phil. Mag., ser. 8, vol. 4, p. 964, 1959.
214 Metallursical Fundamentals
[Ch
ap.
forces to be maintained in the notched bar, it is necessary that no stresses
act normal to the free surfaces of the notch. All the tensile load must be
taken by the metal in the core of the notch. Therefore, a relatively large
mass of unstressed metal exists around a central core of highly stressed
material. The central core tries to contract laterally because of the Pois
son effect, but it is restrained by what amounts
to a hoop of unstressed material around it.
The resistance of the unstressed mass of ma
terial to the deformation of the central core
produces radial and transverse stresses.
The existence of radial and transverse
stresses (triaxial stress state) raises the value
of longitudinal stress at which yielding occurs.
For simplification, consider that yielding oc
curs at a critical shear stress Tc. For an un
notched tension specimen this critical value is
given by
Tc  2
For a notched tension specimen this becomes
Fig. 710. Stress distribution
produced in notched cylinder
under uniaxial loading, ctl =
longitudinal stress; or =
transverse stress; or = ra
dial stress.
Since the critical shear stress for yielding is
the same for both cases, it is apparent from
these equations that the existence of trans
verse stresses requires a higher longitudinal
stress to produce yielding. The entire flow
curve of a notched specimen is raised over
that for an unnotched specimen because of this effect. The amount by
which the flow curve is raised because of the notch can be expressed by a
plasticconstraint factor q.
Plastic constraint differs from elasticstress concentration in a basic
way. From elastic considerations the stress concentration at the root of
a notch can be made extremely high as the radius at the root of
the notch approaches zero. When plastic deformation occurs at the root
of the notch, the elasticstress concentration is reduced to a low value.
However, plastic deformation produces plastic constraint at the root of
the notch. In contrast to elasticstress concentrations, no matter how
sharp the notch the plasticconstraint factor^ cannot exceed a value of
about 3.
' E. Orowan, J. F. Nye, and W. J. Cairns, "Strength and Testing of Materials,"
vol. 1, H. M. Stationery Office, London, 1952.
>cc.
713]
Fracture
215
A third important contribution of a notch is to produce an increase in
the local strain rate. While the notch is still loaded in the elastic region,
the stress at a point near the notch is rapidly increasing with time because
of the sharp gradients. Since stress is proportional to strain, there is a
high local elastic strain rate. When yielding occurs, the plastic flow
tends to relieve the stresses. The stress picture changes from one of
high elastic stresses to a lower plastic constraint, and in so doing a high
plastic strain rate develops near the notch.
71 3. Concept of the Fracture Curve
In earlier chapters it was shown that the flow curve, or the true stress
strain curve, can be considered to represent the stress required to cause
plastic flow at any particular value of plastic strain. In an analogous
manner it was proposed by Ludwik^ that a metal has a fracture stress
BritfleS^—" — ——
Fraclurecurje__^
«i
^r '
!>>l
«..
^
■C: 1
^1
5^1
1
1
1
True strain e
Fig. 71 1 . Schematic drawing of intersec
tion of flow curve and fracture curve ac
cording to the Ludwik theory.
True strain e
Fig. 712. Modification of the Ludwik
theory to include shear and brittle frac
ture curves.
curve which indicates the stress required to cause fracture at any value
of plastic strain. He further suggested that fracture would occur when
the flow curve intersects the fracture curve (Fig. 711). This concept
was widely accepted until after World War II, and several measurements
of the fracture curve were attempted. However, it eventually was real
ized that basic factors in the fracture mechanism of metals prevent a
correct determination of the fracture curve of metals. Since this reali
zation, the idea of the fracture curve has lost much of its popularity.
It is, however, still a useful concept for obtaining a qualitatively correct
picture of the fracture phenomenon if the limitations which will be dis
cussed shortly are recognized. With this in mind, and in view of the
current realization that both shear and cleavage fractures are possible in
metals, it is often found that a separate fracture curve for each mode of
fracture is considered, as shown in Fig. 712. The curves in this figure
1 P. Ludwik, Z. Ver. deut. Ing., vol. 71, pp. 15321538, 1927.
216 Metallurgical Fundamentals [Chap. 7
are for the ordinary tensile fracture of a ductile metal in which a shear
type of fracture takes place. The separation between the two fracture
curves and their relative height will be different for other conditions of
fracture.
In principle, a point on the fracture curve is obtained by plastically
straining a specimen to a given point on the flow curve and then intro
ducing embrittling parameters (low temperature or a notch) so that the
specimen is stressed to failure without added strain. By repeating this
process with different specimens stressed to different values of plastic
strain it would be possible to construct the entire fracture curve. How
ever, since the embrittling effect of a notch is limited to a plasticcon
straint factor of about 3, it is generally more effective to attempt to
resist any further deformation by carrying out the test at a very low
temperature. Actually, with most metals this is not possible since a
slight amount of deformation invariably results on straining at low tem
perature. In view of the evidence that fracture is initiated by plastic
deformation, it would appear that the fracture stress measured by this
technique does not measure the true resistance of the metal to fracture.
Further, the fracture stress for ductile fracture is very difficult to meas
ure accurately because the ductile fracture is initiated at the interior of
the specimen, and the stress distribution is complicated by necking of the
tensile specimen. Therefore, there is no reliable method for determining
the fracture curve of metals. However, this does not prohibit using the
concept of the fracture stress, in a qualitative sense, where it is useful
for describing certain aspects of fracture.
714. Classical Theory of the DuctiletoBrittle Transition
Brittle fracture is promoted by three main factors, (1) a triaxial state
of stress, (2) a low temperature, and (3) a high strain rate. In the previ
ous section it was shown that the presence of a notch provides condition 1
and contributes to condition 3. Temperature has a strong effect on the
basic flow and fracture properties of the metal. For all metals the yield
stress or flow stress increases with decreasing temperature. With fee
metals, where there is no ductiletobrittle transition, the increase in yield
stress on going from room temperature to liquidnitrogen temperature
(— 196°C) is about a factor of 2. In bcc metals, which show a ductile
tobrittle transition, the yield stress increases by a factor of 3 to 8 over
the same temperature range. Figure 75 illustrates the trends in frac
ture stress and yield stress with temperature. It also shows that the
reduction of area at fracture in a tensile specimen drops off rapidly over
a narrow temperature interval. The temperature range at which this
transition occurs is called the transition temperature.
Sec. 714]
Hracture
217
t
The socalled classical theory of the ductiletobrittle transition was
suggested by Davidenkov and Wittman.^ According to this concept,
the existence of a transition temperature is due to the difference in the
way the resistances to shear and cleavage change with temperature. The
relative values of these two parameters determine whether the fracture
will be ductile or brittle. Above the transition temperature the flow
stress is reached before the fracture
stress, while below the transition
temperature the fracture stress is
reached first. Factors which in
crease the critical shear stress for
slip without at the same time raising
the fracture stress will favor brittle
fracture. Decreasing the tempera
ture and increasing the strain rate
both have this effect. In Fig. 713,
the curve marked cto gives the tem
perature dependence of yield stress
for simple tension. The ciu've qao,
where q « 3, is the temperature de
pendence of the yield stress in the
presence of the plastic constraint at
a notch. The curve marked aj is
the fracture strength or cleavage strength as a function of temperature.
In agreement with available data, it is drawn as a less sensitive func
tion of temperature than the yield stress. A transition temperature
occurs when a curve of flow stress intersects the cleavage strength. For
an unnotched tension specimen this occurs at a quite low temperature,
but for a notched test the transition temperature is much closer to room
temperature.
While this picture of the ductiletobrittle transition does not provide
for the structural details embodied in the dislocation theory, it does give
an easily grasped working model of the phenomenon. As originally pro
posed, this classical theory ascribes no major effect to the role of strain
rate; yet recent experiments have indicated that strain rate may be more
important than plastic constraint in producing brittle fracture. Using
sharp cleavage cracks as notches, Felbeck and Orowan^ were unable to
produce cleavage fracture in steel plates unless the crack reached a high
velocity. Extensive plastic deformation was present at the base of the
crack in all cases. These experiments could be interpreted only by con
1 N. N. Davidenkov and F. Wittman, Phys. Tech. Inst. (U.S.S.R.), vol. 4, p. 300,
1937.
^ Felbeck and Orowan, op. cit.
Transition temperature Notcti
simple tension transition temperature
Temperature— >
Fig. 71 3. Schematic description of tran
sition temperature.
218 Metallurgical Fundamentals [Chap. 7
sidering that the jaeld stress is raised to the value of the fracture stress,
not by plastic constraint, but by the effect of high strain rate on increas
ing the yield stress. It is difficult to separate these two effects, and
additional experiments would be very worthwhile. However, it is inter
esting to note that the yield stress of mild steel is very sensitive to
strain rate. Also, the large increase in transition temperature that is
brought about by using a notchedimpact test can be understood on this
basis when it is considered that the strain rate in the impact test is about
10^ times greater than in the ordinary tension test.
71 5. Fracture under Combined Stresses
The phenomenological approach to fracture is concerned with uncover
ing the general macroscopic laws which describe the fracture of metals
under all possible states of stress. This same approach was discussed in
Chap. 3 with regard to the prediction of yielding under complex states
of stress. The problem of determining general laws for the fracture
strength of metals is quite difficult because fracture is so sensitive to
prior plastic straining and temperature. In principle we can conceive of
a threedimensional fracture surface in terms of the three principal stresses
(Ti, 02, and 03. For any combination of principal stresses the metal will
fracture when the limiting surface is reached. Enough experimentation
has been done to realize that the fracture surface cannot be rigid but
must be considered as a flexible membrane which changes shape with
changes in stress and strain history.
Most experimentation in this field has been with biaxial states of stress
where one of the principal stresses is zero. Tubular specimens in which
an axial tensile or compressive load is superimposed on the circumfer
ential stress produced by internal pressure are ordinarily used for this
type of work. For accurate results bulging or necking during the later
stages of the test must be avoided. This makes it difficult to obtain
good data for very ductile metals.
Figure 714 illustrates the fracture criteria which have been most fre
quently proposed for fracture under a biaxial state of stress. The
maximumshearstress criterion and the Von Mises, or distortionenergy,
criterion have already been considered previously in the discussion of
yielding criteria. The maximumnormalstress criterion proposes that
fracture is controlled only by the magnitude of the greatest principal
stress. Available data on ductile metals such as aluminum and mag
nesium alloys^ and steel^ indicate that the maximumshearstress criterion
1 J. E. Dorn, "Fracturing of Metals," American Society for Metals, Metals Park,
Ohio, 1948.
2 E. A. Davis, J. Appl. Mech., vol. 12, pp. A13A24, 1945.
)ec.
716]
Fracture
219
for fracture results in the best agreement. Agreement between experi
ment and theory is not nearly so good as for the case of yielding criteria.
The fracture criterion for a brittle cast iron^ is shown in Fig. 715. Note '
that the normal stress criterion is followed in the tensiontension region
and that the fracture strength increases significantly as one of the princi
pal stresses becomes compressive. Two theories^ which consider the
+02
Maximum
normal
stress^
^ ' ~^^
' /7
\
Von Mises /y^
I
criterion — V/
1
//
/
a, /
/ +^1
I
1
^l\/faximum
//
//
/ /
1
1
shear
1
I
stress
//
/y^
a2
Fig. 714. Proposed fracture criteria for
biaxial state of stress in ductile metals.
Fig. 715. Biaxial fracture criterion for
brittle cast iron.
stress concentration of graphite flakes in cast iron are in good agreement
with the fracture data. These data are also in substantial agreement
with the fracture curve predicted from Griffith's theory of brittle fracture.
716. Effect of High Hydrostatic Pressure on Fracture
Bridgman's work^ on the effect of superimposed hydrostatic pressure
on the fracture characteristics of metals has produced many interesting
results. His results have also shown that fracture is a complex phe
nomenon which in many cases cannot be described by the simple criteria
of the previous section.
Bridgman tested metal specimens in such a way that hydrostatic pres
sures of up to 450,000 psi were superimposed on an axial tensile stress.
These extreme conditions produced a very great increase in the ductility
at fracture. The strain at the location of fracture was as much as 300
times greater when mild steel was fractured with superimposed hydro
1 W. R. Clough and M. E. Shank, Trans. A&M, vol. 49, pp. 241262, 1957.
2L. F. Coffin, Jr., J. Appl. Mech., vol. 17, p. 233, 1950.
3 J. C. Fisher, ASTM Bull 181, p. 74, April, 1952.
^ P. W. Bridgman, "Studies in Large Plastic Flow and Fracture," McGrawHill
Book Company, Inc., New York, 1952.
220 Metallurgical Fundamentals [Chap. 7
static pressure than when tested with simple uniaxial loading. Materials
which are completely brittle under ordinary conditions, like limestone or
rock salt, actually necked down when pulled in tension with superimposed
hydrostatic pressure. It was also found that, if a tensile specimen was
loaded with superimposed pressure to a point short of fracture and then
tested at atmospheric pressure, it required further deformation before
fracturing, even if the elongation under pressure was greater than the
metal could v/ithstand when ordinarily tested at atmospheric pressure.
Further, the amount of deformation required to produce fracture after
removal of the hydrostatic pressure increases with an increase in the
magnitude of the pressure. These facts indicate that, in general, frac
ture is not determined completely by the instantaneous state of stress or
strain. Bridgman was able to find no simple stress function which
described his results.
BIBLIOGRAPHY
Averbach, B. L., D. K. Felbeck, G. T. Hahn, and D. A. Thomas (eds.): "Fracture,"
Technology Press and John Wiley & Sons, Inc., New York, 1959.
Barrett, C. S. : Metallurgy at Low Temperatures, Campbell Memorial Lecture, 1956,
Trans. ASM, vol. 49, pp. 53117, 1957.
"Fracturing of Metals," American Society for Metals, Metals Park, Ohio, 1948.
Orowan, E.: Fracture and Strength of SoUds, Repts. Progr. in Phys., vol. 12, pp. 185
232, 1949.
Parker, E. R. : "Brittle Behavior of Engineering Structures," John Wiley & Sons,
Inc., New York, 1957.
Patch, N. J.: The Fracture of Metals, in "Progress in Metal Physics," vol. 5, Perga^
mon Press, Ltd., London, 1954.
Stroh, A. N.: Advances in Phys., vol. 6, pp. 418465, 1957.
Chapter 8
INTERNAL FRICTION
81 . Introduction
The ability of a vibrating solid which is completely isolated from its
surroundings to convert its mechanical energy of vibration into heat is
called internal friction, or damping capacity. The former term is pre
ferred by physicists, and the latter is generally used in engineering. If
metals behaved as perfectly elastic materials at stresses below the nominal
elastic limit, there would be no internal friction. However, the fact that
damping effects can be observed at stress levels far below the macroscopic
elastic limit indicates that metals have a very low true elastic limit, if,
indeed, one exists at all. Internalfriction, or damping, effects corre
spond to a phase lag between the applied stress and the resulting strain.
This may be due simply to plastic deformation at a high stress level,
or at low stress levels it may be due to thermal, magnetic, or atomic
rearrangements.
An important division of the field of nonelastic behavior is called
anelasticity . This subject is concerned with internalfriction effects which
are independent of the amplitude of vibration. Anelastic behavior can
be due to thermal diffusion, atomic diffusion, stress relaxation across
grain boundaries, stressinduced ordering, and magnetic interactions.
Certain static effects such as the elasticaftereffect are concerned with
anelastic behavior. Internal friction resulting from cold work is strongly
amplitudedependent and, therefore, is not an anelastic phenomenon.
Much of our present knowledge of the mechanisms which contribute to
anelasticity is due to Zener^ and his coworkers.
Studies of internal friction are primarily concerned with using damping
as a tool for studying internal structure and atom movements in solids.
The method has provided information on diffusion, ordering, and solu
bilities of interstitial elements and has been used for estimating the den
sity of dislocations. The vibration amplitudes employed in this type of
1 C. Zener, "Elasticity and Anelasticity," University of Chicago Press, Chicago;
1948.
221
222 Metallurgical Fundamentals [Chap. 8
work are usually quite small, and the stresses are very low. Another
aspect of this field is the determination of engineering data on the dissi
pation of energy in vibrating members. This work is usually concerned
with determining the damping capacity of a material at the relatively
large amplitudes encountered in engineering practice.
Internal friction is measured by a number of techniques.^ The sim
plest device is a torsional pendulum for use in the lowfrequency region
around 1 cps. For higherfrequency measurements the specimen is
excited by an electromagnetic drive, a piezoelectric crystal, or ultrasonic
energy.
82. Phenomenological Description of Internal Friction
For energy to be dissipated by internal friction, the strain must lag
behind the applied stress. The phase angle, or lag angle, a can be used
as a measure of internal friction.
a~4^ (81)
where eg' = nonelastic strain component 90° out of phase with stress
ei = elastic strain in phase with stress
Internal friction is frequently measured by a system which is set into
motion with a certain amplitude ^o and then allowed to decay freely.
The amplitude at any time. At, can be expressed by an equation
At = Aoexp(/30 (82)
where /3 is the attenuation coefficient. The most common way of defining
internal friction or damping capacity is with the logarithmic decrement 5.
The logarithmic decrement is the logarithm of the ratio of successive
amplitudes.
h = In 4^ (83)
If the internal friction is independent of amplitude, a plot of In A versus
the number of cycles of vibration will be linear and the slope of the curve
is the decrement. If the damping is amplitudedependent, the decre
ment is given by the slope of the curve at a chosen amplitude. The
logarithmic decrement is related to the lag angle by
5 = Tza (84)
For a condition of forced vibration in which the specimen is driven at
1 C. Wert, "Modern Research Techniques in Physical Metallurgy," pp. 225250,
American Society for Metals, Metals Park, Ohio, 1953.
Sec. 821
Internal Friction
223
a constant amplitude a measure of internal friction is the fractional
decrease in vibrational energy per cycle. Vibrational energy is propor
tional to the square of the amplitude, so that the logarithmic decrement
can be expressed by
AW
i = 2W (85)
where AW is the energy lost per cycle and W is the vibrational energy
at the start of the cycle. In a
forcedvibration type of experiment
it is customary to determine a reso
nance curve such as that of Fig. 81.
The logarithmic decrement for a
resonance curve is given approxi
mately by
8 =
7r(bandwidth) 7r(/2 — /i)
fr
fr
(86)
A measure of internal friction that
is often used is the Q, where
Q = ir/8. Since in electricalcircuit
theory the reciprocal of this value
is called the Q of the circuit, the symbol Q^^ has been adopted as a
measure of internal friction.
Frequency
Fig. 81. Resonance curve.
Q' =
hh
fr
(87)
Under conditions of cyclic excitation the dynamic elastic modulus will
be greater than the static elastic modulus because of nonelastic internal
friction. The modulus under dynamic conditions is frequently termed
the unrelaxed elastic modulus Eu, while the static modulus is called the
relaxed modulus Er. The unrelaxed modulus is given by
Eu
01
ei'
(88)
where ei^ is the elastic and ei^ is the plastic strain component in phase
with the stress. The fact that the dynamic modulus is larger than the
static elastic modulus is called the AE effect.
A number of models have been proposed to describe the nonelastic
behavior of materials. The models suggested by Voight^ and Maxwell^
' W. Voight, Ann. Physik, vol. 47, p. 671, 1892.
2 J. C. Maxwell, Phil. Mag., vol. 35, p. 134, 18G8.
224
Metallurgical Fundamentals
[Chap. 8
are frequently mentioned. Both models consider that the material has
an elastic component coupled with a viscous component. The behavior
of a material, with the properties attributed to it by the theory, can be
duplicated by a mechanical model composed of springs (elastic com
ponent) and dashpots (viscous component). Figure 82 illustrates the
composition of a Voight and Maxwell solid, together with the equations
which the models predict. For real metals the frequency dependence
of internal friction does not agree with the equations predicted by the
models. Further, the models do not account for the dependence of
dynamic modulus on internal friction, which is observed with real metals.
Various modifications of the models have been useful in studying the
mechanical properties of polymers, but they are of limited usefulness in
dealing with metals.
Voight solid
Maxwell solid
cr= F^e + 7j€
Eu V
7
Eu
oc log angle
^= Zvf
7 = coefficient of
viscos
ty
Eu = dynomic mod
jIus
Time
Time — >
Fig. 82. Spring and dashpot models of Fig. 83. Time dependence of stress and
Voight and Maxwell solids. strain for anelastic material, showing
elastic aftereffect.
83. Anelasticity
A nonelastic body is said to behave anelastically when the stress and
strain are not singlevalued functions of each other and the internal fric
tion is independent of amplitude. One manifestation of this is the elastic
aftereffect. Consider a metal which is subjected to a constant stress at a
level well below the conventional elastic limit (Fig. 83). The strains
involved may be of the order of 10~^, so that very sensitive measurements
are required. After an initial strain the metal will gradually creep until
the strain reaches an essentially constant value. This can be observed
Sec. 83]
Internal Friction
225
in many metals at room temperature, although the effect is greater at
higher temperatures. When the stress is removed, the strain will
decrease but there will be a certain amount which remains and slowly
decreases with time, approaching its original value. This time depend
ence of strain on loading and unloading has been called the elastic
aftereffect.
In considering the stressstrain relationship for an anelastic material
it is apparent that a constant linear relationship between these two fac
tors will not adequately describe
the situation. A realistic relation
ship is obtained by equating the
stress and its first derivative with
respect to time to the strain and
the strain rate.
aid \ aib = hie { 626 (89)
Time
A material which obeys this type
of equation is known as a standard
linear solid. The mechanical model
for this material is shown in Fig.
84. Note that the time depend
ence of strain closely duplicates the
behavior of a material with an elastic aftereffect. The general equation
for a standard linear solid can be rewritten in terms of three independent
constants.
Time
Fig. 84. Mechanical model of standard
linear solid and associated time depend
ence of stress and strain.
(T + T,& = ER(e + T^e)
(810)
where t^ = time of relaxation of stress for constant strain
Ta = time of relaxation of strain for constant stress
Er = relaxed elastic modulus
The relationship between the relaxation times and the relaxed and
unrelaxed modulus is given by
Er
Eu
(811)
A dimensionless combination of elastic constants, called the relaxation
strength, is a measure of the total relaxation
E. =
Eu — Ej
(812)
VEuEr
For a standard linear solid there is only a single relaxation time
226 Metallurgical Fundamentals [Chap. 8
T = (r, + T<,)/2. The lag angle, on the basis of this model, is given by
the following equation:^
a = E, .. 2'' 2 2 (813)
where co = 2x/is the angular frequency of vibration. Equation (813) is
symmetrical in both co and r and has a maximum when cor = 1. There
fore, for a material which behaves like an anelastic standard linear solid^
an internalfriction peak will occur at an angular frequency which is the
reciprocal of the relaxation time of the process causing the relaxation.
It is often difficult experimentally to vary the angular frequency by a
factor much greater than 100. Therefore, it is usually easier to deter
mine the relaxation spectrum by holding co constant and varying the
relaxation time r. In many materials, including metals, r varies expo
nentially with temperature so that
r = TO exp ^ (814)
Therefore, to determine the relaxation spectrum, all that is necessary is
to measure a as a function of temperature for constant angular frequency.
Internalfriction measurements are well suited for studying the dif
fusion of interstitial atoms in bcc metals. Relaxation peaks arise owing
to diffusion of interstitial atoms to minimumenergy positions in the
stress fields of the dislocations. For a given frequency the relaxation
time is expressed by r = 1/co, and the peak occurs at a temperature Ti.
At another value of frequency the relaxation peak will occur at a tem
perature To. From the temperature dependence of relaxation time [Eq.
(814)] the activation energy Aiif can be determined.
For a given relaxation time the diffusion coefficient of the interstitial
atoms is given by
D = ^ (816)
dor
where ao is the interatomic spacing. The temperature dependence of D
is given by
D = D, exp  1^ (817)
1 A. S. Nowick, Internal Friction in Metals, in "ProgjRss in Metal Physics," vol. 4.
pp. 1516, Pergamon Press, Ltd., London, 1953.
Sec. 85] Internal Friction 227
84. Relaxation Spectrum
A number of relaxation processes with different relaxation times can
occur in metals. Each will occur in a different frequency region, so that
a number of internalfriction peaks can be found if a wide range of fre
quency is investigated. Provided that the peaks are sufficiently sepa
rated, the behavior of the metal in the region of the peak can be expressed
by Eq. (810) with suitably determined constants. This variation of
internal friction with frequency can be considered as a relaxation spec
trum which is characteristic of a particular material.
The application of stress to a substitutional solid solution can produce
ordering in an otherwise random distribution of atoms. An alternating
stress can give rise to relaxation between pairs of solute atoms.
A large and broad internalfriction peak is produced in polycrystalline
specimens by the relaxation of shear stress across grain boundaries.
Work in this area has led to the conclusion that grain boundaries behave
in some ways like a viscous material. This interesting aspect of internal
friction is discussed in more detail in the next section.
The movement of lowenergy twin boundaries due to stress is believed
to produce relaxation effects.' This type of deformation is also responsi
ble for anelastic effects found in conjunction with domainboundary
movement in ferromagnetic materials. Since twin interfaces are crys
tallographically coherent boundaries, the internal friction cannot be due
to the viscous slip associated with incoherent boundaries.
The relaxation peak due to preferential ordering of interstitial atoms
in the lattice from an applied stress is one of the bestunderstood relax
ation processes. Studies of this relaxation process have provided data
on the solubility and diffusion of interstitial atoms. This type of internal
friction is considered in Sec. 86. Relaxation produced by thermal fluctu
ations will be considered in Sec. 87.
85. Grainboundary Relaxation
An important source of internal friction in metals is stress relaxation
along grain boundaries. Ke first demonstrated the strong internal
friction peak due to grainboundary relaxation by experiments on high
purity aluminum wires. At the low torsional strains used in this work the
strain was completely recoverable, and all internalfriction effects were
independent of amplitude. Ke found that a broad peak occurred in the
region of 300°C in polycrystalline aluminum, while no internalfriction
1 F. T. Worrell, J. Appl. Phys., vol. 19, p. 929, 1948, vol. 22, p. 1257, 1951.
2 T. S. Ke, Phys. Rev., vol. 71, p. 533, vol. 72, p. 41, 1947.
228
Metallurgical Fundamentals
[Chap. 8
peak was observed in aluminum single crystals (Fig. 85). In addition,
measurements of the modulus (which is proportional to the square of
frequency) at different temperatures showed a sharp drop for the poly
crystalline specimen which was not found with the singlecrystal specimen
(Fig. 86). This behavior is consistent with the assumption that grain
boundaries behave to a certain extent in a viscous manner at elevated
temperatures.
U.I u
0.08
Or
i 0.06
§0.04
CZ
0.02
1
1
1
1

/
\ Polycrystalline
\ aluminum

/
\


/
^
9=r^^
U
< 'T
"single crystal"
aluminum) ^ , __
1 1
100 200 300 400
Temperature of measurement, °C
500
Fig. 85. Variation of internal friction with temperature for polycrystalline and single
crystal specimens of aluminum. {T . S. Ke, Phys. Rev., vol. 71, p. 533, 1947.)
100 200 300 400
Temperature of measurement , °C
500
Fig. 86. Variation of modulus (p) with temperature for polycrystalline and single
crystal aluminum. {T. S. Ke, Phys. Rev., vol. 71, p. 533, 1947.)
Sec. 87J Internal Friction 229
86. The Snock Effect
Internal friction resulting from preferential ordering of interstitial
atoms under applied stress was first explained by Snoek' and is known as
the Snoek effect. This type of relaxation has been most extensively
studied in iron containing small amounts of either carbon or nitrogen in
solid solution. Interstitial carbon atoms in bcc iron occupy the octa
hedral holes in the lattice. Even though no external forces are applied,
the crystal will have tetragonal symmetry because of the distortion pro
duced by the interstitial atoms. As long as no stress is applied, the
distribution of atoms among the octahedral sites is random and the
tetragonal axes of the unit cells are randomly oriented with respect to the
specimen axes. However, if a stress is applied along the y axis, the inter
stitial atoms will migrate to octahedral positions which tend to give a
preferred alignment in the y direction. When the stress is removed, the
atoms will migrate toward a random distribution. Under the oscillating
stresses imposed by an internalfriction apparatus the interstitial atoms
will be in continuous motion, either tending toward or tending away from
a preferred orientation. A strong relaxation peak results. A similar but
weaker relaxation peak can be observed due to shortrange order in
substitutional solid solutions.
87. Thermoelastic Internal Friction
The thermal and the mechanical behavior of materials are interrelated.
The application of a small stress to a metal will produce an instantaneous
strain, and this strain will be accompanied by a small change in tem
perature. An extension of the specimen will result in a decrease in tem
perature, while a contraction produces a temperature rise. This behavior
is called the thermoelastic effect. If the applied stress is not uniform
throughout the specimen, a temperature gradient will be set up and addi
tional nonelastic strain will result. If the nonuniform stress varies
periodically with time, a fluctuating temperature gradient is produced.
When the stress fluctuations occur at a very high frequency, so that there
is not time for appreciable heat flow to take place during a stress cycle, the
process is adiabatic. No energy loss or damping occurs under adiabatic
conditions. On the other hand, at very low frequencies there is adequate
time for heat flow, and an equilibrium temperature is maintained in the
specimen. This is an isothermal process, and no energy or heat is lost.
In the region of intermediate frequencies the conversion of energy into
heat is not reversible, and internalfriction effects are observed.
Nonuniform stress can result in macroscopic thermal currents which
1 J. Snoek, Physica, vol. 6, p. 591, 1939, vol. 8, p. 711, 1941, vol. 9, p. 862, 1942.
230 Metallurgical Fundamentals [Chap. 8
produce internalfriction peaks. A rectangular bar which is vibrated
transversely behaves like a standard linear solid (single relaxation time).
The compression side of the specimen will increase in temperature, and
the tension side will undergo a decrease in temperature. Therefore, an
alternating temperature gradient is produced across the thickness of the
bar. A relaxation process occurs, provided that the frequency is such
that there is enough time for thermal currents to flow back and forth
and effect a partial neutralization of the temperature gradient. Zener^
has shown that the relaxation time is
(818)
irWt
where h = thickness of specimen
Dt = thermaldiffusion constant
= thermal conductivity/ (specific heat) (density)
The frequency at which this relaxation peak occurs can be determined
from the relationship w^ = 1. For specimens of ordinary thickness, the
peak would occur in the region of 1 to 100 cps. It is theoretically possi
ble for a specimen vibrated in a longitudinal mode to show relaxation
from macroscopic thermal currents. However, the frequency region
where the peak would occur would be of the order 10^" to 10^^ cps, which
is well beyond the range of normal observations. No relaxation from
macroscopic thermal currents occurs in a specimen subjected to torsional
vibration, because shearing stresses are not accompanied by a change in
temperature.
A poly crystalline specimen which is subjected to completely uniform
stress can show relaxation due to intergranular thermal currents arising
from the fluctuations in stress from grain to grain. The localized stress
differences from grain to grain are due to the elastic anisotropy of indi
vidual grains. The relaxation peak due to intergranular thermal cur
rents will not occur at a sharp frequency and, therefore, represent a single
relaxation time. The frequency at which the relaxation will occur is
related to the grain size of the metal. Internal friction due to inter
granular thermal currents can occur for all types of stressing. It is
important that effects from this source be considered in experiments
where the prime interest is in damping from other sources.
88. Dislocation Damping
The internal friction of metals is quite sensitive to plastic deformation.
The effects are very complex and depend on variables such as the amount
of plastic deformation, the method by which the deformation was intro
1 C. Zener, Phys. Rev., vol. 52, p. 230, 1937.
Sec. 88] Internal Friction 231
duced into the metal, the purity of the metal, the frequency of vibration,
and the time between the deformation and the measurement of internal
friction. Read^ demonstrated that internal friction arising from cold
work is strongly amplitudedependent, even for strain amplitudes as
small as 10^^.
A freshly coldworked metal has a relatively high internal friction which
anneals out very rapidly at temperatures well below those required for
recrystallization. The high damping is also accompanied by a decrease
in the dynamic modulus. As the internal friction anneals out, the
dynamic modulus returns to its steadystate value. The decrease in
modulus due to cold work which can be eliminated by annealing at rela
tively low temperatures is called the modulus defect, or the Koster effect.^
Mott has proposed a dislocation modeP for the modulus defect which is
based on the bowing out under stress of a network of dislocation lines
anchored at nodes and impurities. The theory predicts that the modulus
defect is proportional to the products of the dislocation length per cubic
centimeter and the square of the effective loop length of a dislocation
segment,
^ cc NU (819)
In a coldworked metal typical values of A^" ^^ 10' and L c^ 10~^ cm would
lead to values of I\E/E of 10 per cent, in agreement with observed results.
The dislocation mechanism for the internalfriction effects observed in
coldworked metals is not well established. The theory due to Koehler*
and Granato and Liicke^ assumes that amplitudedependent internal fric
tion is due to a stressstrain hysteresis arising from the irreversibility of
dislocation lines breaking away from pinning impurity atoms. However,
amplitudeindependent internal friction is assumed to result from a
viscouslike damping force acting on the bowedout segments of the dis
location lines.
The only relaxation process which gives an internal peak that is
definitely ascribable to dislocations is the Bordoni peak^ found in fee
metals at very low temperatures in the region of 30 to 100°K. There
are indications that the Bordoni peak is due to some intrinsic property
of dislocations and is not involved with the interaction of dislocations
with impurity atoms and other dislocations.
1 T. A. Read, Trans. AIME, vol. 143, p. 30, 1941.
2 W. Koster, Z. Metallk., vol. 32, p. 282, 1940.
3 N. F. Mott, Phil. Mag., vol. 43, p. 1151, 1952.
^ J. S. Koehler, "Imperfections in Nearly Perfect Crystals," John Wiley & Sons,
Inc., New York, 1953.
5 A. Granato and K. Lucke, J. Appl. Phys., vol. 27, p. 583, 1956.
® P. G. Bordoni, Nuovo cimento, vol. 7, ser. 9, suppl. 2, p. 144, 1950
232 Metallurgical Fundamentals [Chap. 8
89. Dampins Capacity
This section is concerned with the engineering aspects of internal fric
tion. The damping capacity of structures and machine elements is con
cerned with the internal friction of materials at strain amplitudes and
stresses which are much greater than the values usually considered in
internalfriction experiments. A high damping capacity is of practical
engineering importance in limiting the amplitude of vibration at reso
nance conditions and thereby reducing the likelihood of fatigue failure.
Turbine blades, crankshafts, and aircraft propellers are typical applica
tions where damping capacity is important.
Damping capacity can be defined as the amount of work dissipated
into heat per unit volume of material per cycle of completely reversed
stress. The damping properties of materials are frequently expressed in
terms of the logarithmic decrement 6 or the specific damping capacity \l/.
^ = 2. = I^L^ (820)
where \l/ = specific damping capacity
6 = logarithmic decrement [see Eqs. (83) and (85)]
Ai — amplitude of vibration of first cycle
An = amplitude of vibration of nth cycle
N = number of cycles from Ai to An
Values of these damping parameters depend not only on the condition of
the material but also on the shape and stress distribution of the speci
mens. Since these conditions are often not specified, there is considera
ble variation and contradiction in the published literature^ on the damp
ing properties of materials. The proposal has been made to express the
engineering damping properties of materials hy the specific damping
energy. This quantity represents the area inside a stressstrain hysteresis
loop under uniform stress conditions and is a true material property.
Methods of converting logarithmic decrement and damping capacity to
specific damping energy have been published.^
Engineering dampingcapacity measurements are not very dependent
upon frequency of vibration. They are, however, strongly dependent on
the stress or strain amplitude. Specific damping energy is approximately
a power function of stress level, with the exponent varying between 2 and
' L. J. Demer, Bibliography of the Material Damping Field, WADC Tech. Rept.
56180, June, 1956; available from Office of Technical Services.
2 E. R. Podnieks and B. J. Lazan, Analytical Methods for Determining Specific
Damping Energy Considering Stress Distribution, WADC Tech. Rept. 5644, June,
1957.
Sec. 89]
Internal Friction
233
3 for most materials. The damping behavior is a function of the number
of reversed stress cycles. Generally, the damping capacity increases
with number of cycles of stress reversal, the magnitude of the effect
increasing with stress level. The damping capacity for a given metal
and test condition depends on the type of stress system, i.e., whether
tested in torsion or tension. This is the result of differences in stress
distribution produced by different methods. A number of attempts
have been made to relate damping behavior with other properties such as
fatigue strength and notch sensitivity. While in certain cases it appears
Table 81
Damping Capacity of Some Engineering MATERiALsf
Material
Carbon steel (0.1 % C)
NiCr steel — quenched and tempered.
12% Cr stainless steel
188 stainless steel
Cast iron
Yellow brass
Specific damping capacity
at various stress levels APF/PT
4,500 psi 6,700 psi 11,200 psi
2.28
0.38
8.0
0.76
28.0
0.50
2.78
0.49
8.0
1.16
40.0
0.86
4.16
0.70
8.0
3.8
t S. L. Hoyt, "Metal Data," rev. ed., Reinhold Publishing Corporation, New York,
1952.
that high damping capacity correlates with a low notch sensitivity,
there is no general relationship between these properties. Furthermore,
there is no general relationship between damping capacity and fatigue
limit.
Table 81 lists some values of damping capacity for a number of
engineering materials at several stress levels. Cast iron has one of the
highest damping capacities of these materials. This is attributed to
energy losses in the graphite flakes. One important contribution to
damping in many alloys used for turbineblade applications comes from
the motion of ferromagnetic domain walls. This has been demonstrated^
by the fact that a ferromagnetic alloy which showed high damping had
much decreased damping capacity when tested in a magnetic field. The
lower damping in the magnetic field can be attributed to the fact that the
domains are lined up in the direction of the field and cannot move freely
under stress.
1 A. W. Cochardt, Trans. AIME, vol. 206, pp. 12951298, 1956.
234 Metallurgical Fundamentals [Chap. 8
BIBLIOGRAPHY
Entwistle, K. M. : The Damping Capacity of Metals, in B. Chalmers and A. G. Quarrell
(eds.), "The Physical Examination of Metals," 2d ed., Edward Arnold &
Co., London, 1960.
Niblett, D. H., and J. Wilks: Dislocation Damping in Metals Advances inPhys.,
vol. 9, pp. 188, 1960.
Nowick, A. S.: Internal Friction in Metals, in "Progress in Metal Physics," vol. 4,
Pergamon Press, Ltd., London, 1953.
Zener, C: "Elasticity and Anelasticity of Metals," University of Chicago Press,
Chicago, 1948.
Part Three
APPLICATIONS TO MATERIALS TESTING
Chapter 9
THE TENSION TEST
91 . Engineerins StressStrain Curve
The engineering tension test is widely used to provide basic design
information on the strength of materials and as an acceptance test for
the specification of materials. In
the tension test' a specimen is
subjected to a continually increas
ing uniaxial tensile force while si
multaneous observations are made
of the elongation of the specimen.
An engineering stressstrain curve
is constructed from the loadelonga
tion measurements (Fig. 91 ) . The
significant points on the engineer
ing stressstrain curve have already
been considered in Sec. 15, while
the appearance of a yield point in
the stressstrain curve was covered
in Sec. 55. The stress used in this
stressstrain curve is the average
longitudinal stress in the tensile
specimen. It is obtained by dividing the load by the original area of
the cross section of the specimen.
Fig. 91,
curve.
Conventional strain e
The engineering stressstrain
P_
(91)
The strain used for the engineering stressstrain curve is the average
1 H. E. Davis, G. E. Troxell, and C. T. Wiskocil, "The Testing and Inspection of
Engineering Materials," 2d ed., chaps. 24, McGrawHill Book Company, Inc., New
York 1955.
237
238 Applications to Materials Testing [Chap. 9
linear strain, which is obtained by dividing the elongation of the gage
length of the specimen, 8, by its original length.
Since both the stress and the strain are obtained by dividing the load and
elongation by constant factors, the loadelongation curve will have the
same shape as the engineering stressstrain curve. The two curves are
frequently used interchangeably.
The shape and magnitude of the stressstrain curve of a metal will
depend on its composition, heat treatment, prior history of plastic defor
mation, and the strain rate, temperature, and state of stress imposed
during the testing. The parameters which are used to describe the stress
strain curve of a metal are the tensile strength, yield strength or yield
point, per cent elongation, and reduction of area. The first two are strength
parameters; the last two indicate ductility.
Tensile Strength r
The tensile strength, or ultimate tensile strength (UTS), is the maxi
mum load divided by the original crosssectional area of the specimen.
au = ^ (93)
The tensile strength is the value most often quoted from the results of a
tension test; yet in reality it is a value of little fundamental significance
with regard to the strength of a metal. For ductile metals the tensile
strength should be regarded as a measure of the maximum load which a
metal can withstand under the very restrictive conditions of uniaxial
loading. It will be shown that this value bears little relation to the useful
strength of the metal under the more complex conditions of stress which
are usually encountered. For many years it was customary to base the
strength of members on the tensile strength, suitably reduced by a factor
of safety. The current trend is to the more rational approach of basing
the static design of ductile metals on the yield strength. However,
because of the long practice of using the tensile strength to determine
the strength of materials, it has become a very familiar property, and as
such it is a very useful identification of a material in the same sense that
the chemical composition serves to identify a metal or alloy. Further,
because the tensile strength is easy to determine and is a quite repro
ducible property, it is useful for the purposes of specifications and for
quality control of a product. Extensive empirical correlations between
tensile strength and properties such as hardness and fatigue strength are
Sec. 91] The Tension Test 239
often quite useful. For brittle materials, the tensile strength is a valid
criterion for design.
Yield Strength
The yield strength is the load corresponding to a small specified plastic
strain divided by the original crosssectional area of the specimen.
^6=0.002 ,p, ..
c^o = — 2 (94)
Because of the practical difficulties of measuring the elastic limit or pro
portional limit, the yield strength and yield point are the preferred
engineering parameters for expressing the start of plastic deformation.
When the design of a ductile metal requires that plastic deformation
be prevented, the yield strength is the appropriate criterion of the
strength of the metal. An important feature of the yield strength is
that the value determined from the tension test can be used to predict
the conditions for static yielding under other, more complex conditions of
stress by means of the distortionenergy yielding criterion (Sec. 34).
An example of this is the determination of the elasticbreakdown pressure
of thickwall tubes subjected to internal pressure from the results of a
tension test.^ The yield strength and yield point are more sensitive
than the tensile strength to differences in heat treatment and method of
testing.
Percentage Elongation
The percentage elongation is the ratio of the increase in the length of
the gage section of the specimen to its original length, expressed in per
cent.
% elongation = ^ = e/ (95)
where L/ = gage length at fracture
Lo = original gage length
€f = conventional strain at fracture
The numerator in Eq. (95) is simply the total measured elongation of
the specimen. This value is influenced by the deformation during the
necking of the specimen, and hence the value of per cent elongation
depends somewhat on the specimen gage length. The elongation of the
specimen is uniform along the gage length up to the point of maximum
load. Beyond this point necking begins, and the deformation is no longer
uniform along the length of the specimen. This uniform strain is of more
fundamental importance than total strain to fracture, and it is also of
1 J. H. Faupel, Trans. ASME, vol. 78, pp. 10311064, 1956.
240 Applications to Materials Testing [Chap. 9
some practical use in predicting the formability of sheet metal. How
ever, the uniform elongation is not usually determined in a routine ten
sion test, so that, unless specifically stated, the percentage elongation ifi
always based on the total elongation. It is determined by putting the
broken tensile specimen together and measuring the change in gage
length. The original gage length should always be given in reporting
percentage elongation values.
Reduction of Area
The percentage reduction of area is the ratio of the decrease in the cross
sectional area of the tensile specimen after fracture to the original area,
expressed in per cent.
An — At
Reduction of area = q — ^ (96)
The determination of the reduction of area in thin sheet specimens is
difficult, and for this reason it is usually not measured in this type of
specimen. For thicker, flat, rectangular tensile specimens, the area after
fracture may be approximated by
A = ^{a^2d) (97)
where h, = width of specimen
a = thickness at center of specimen
d = thickness at ends of cross section of specimen
The elongation and reduction of area are usually not directly useful to
the designer. There appear to be no quantitative methods for determin
ing the minimum elongation or reduction of area which a material must
have for a particular design application. However, a qualitative indica
tion of formability of a metal can sometimes be obtained from these
values. A high reduction of area indicates the ability of the metal to
deform extensively without fracture (see Prob. 9.4).
The reduction of area is the most structuresensitive parameter
that is measured in the tension test. Therefore, its most important
aspect is that it is used as an indication of material quality. A decrease
in reduction of area from a specified level for which experience has
shown that good service performance will result is a warning that quality
is substandard.
Modulus of Elasticity
The slope of the initial linear portion of the stressstrain curve is the
modulus of elasticity, or Young's modulus. The modulus of elasticity
is a measure of the stiffness of the material. The greater the modulus.
Sec. 91]
TheT
ension
Test
241
the smaller the elastic strain resulting from the application of a given
stress. Since the modulus of elasticity is needed for computing deflec
tions of beams and other members, it is an important design value.
The modulus of elasticity is determined by the binding forces between
atoms. Since these forces cannot be changed without changing the basic
nature of the material, it follows that the modulus of elasticity is one of
the most structureinsensitive of the mechanical properties. It is only
slightly affected by alloying additions, heat treatment, or cold work.^
However, increasing the temperature decreases the modulus of elasticity.
The modulus is usually measured at elevated temperatures by a dynamic
method^ which measures the mode and period of vibration of a metal
specimen. Typical values of the modulus of elasticity for common engi
neering metals at different temperatures are given in Table 91.
Table 91
Typical Values of Modulus of Elasticity at
Different Temperatures
Material
Modulus of elasticity, psi X 10~«
Room temp.
400°F
800°F
1000°F
1200°F
Carbon steel
30.0
28.0
16.5
10.5
27.0
25.5
14.0
9.5
22.5
23.0
10.7
7.8
19.5
22.5
10.1
18
Austenitic stainless steel
Titanium alloys
21.0
Aluminum alloys
Resilience
The ability of a material to absorb energy when deformed elastically
and to return it when unloaded is called resilience. This is usually meas
ured by the modulus of resilience, which is the strain energy per unit vol
ume required to stress the material from zero stress to the yield stress oq.
Referring to Eq. (257), the strain energy per unit volume for uniaxial
tension is
From the above definition the modulus of resilience is
TT 1/ 1 / '''0 ^0"
(98)
This equation indicates that the ideal material for resisting energy loads
in applications where the material must not undergo permanent distor
tion, such as mechanical springs, is one having a high yield stress and a
' D. J. Mack, Trans. AIME, vol. 166, pp. 6885, 1946.
2 C. W. Andrews, Metal Progr., vol. 58, pp. 8589, 96, 98, 100, 1950.
242 Applications to Materials Testing
[Chap. 9
low modulus of elasticity. Table 92 gives some values of modulus of
resilience for different materials.
Table 92
Modulus of Resilience for Various Materials
Modulus of
Material
E, psi
oQ, psi
resilience Ur,
in.lb/in.3
Mediumcarbon steel
30 X 108
30 X 108
45,000
140,000
33 7
Highcarbon spring steel
320
Duraluminum
10.5 X 108
18,000
17
Copper
16 X 108
4,000
5.3
Rubber
150
300
300
///g^ carbon spring steel
Toughness
The toughness of a material is its ability to absorb energy in the plastic
range. The ability to withstand occasional stresses above the yield stress
without fracturing is particularly desirable in parts such as freightcar
couplings, gears, chains, and crane hooks. Toughness is a commonly
used concept which is difficult to pin down and define. One way of
looking at toughness is to consider that it is the total area under the
stressstrain curve. This area is an indication of the amount of work
per unit volume which can be done on the material without causing it to
rupture. Figure 92 shows the
stressstrain curves for high and
lowtoughness materials. The high
carbon spring steel has a higher
yield strength and tensile strength
than the mediumcarbon structur
al steel. However, the structural
steel is more ductile and has a
greater total elongation. The total
area under the stressstrain curve
is greater for the structural steel,
and therefore it is a tougher mate
rial. This illustrates that tough
ness is a parameter which com
The crosshatched regions in Fig. 92
Because of its higher
Stra,: e
Fig. 92. Comparison of stressstrain
curves for high and lowtoughness mate
rials.
prises both strength and ductility
indicate the modulus of resilience for each steel
yield strength, the spring steel has the greater resilience.
Several mathematical approximations for the area under the stress
strain curve have been suggested. For ductile metals which have a
Sec. 92J The Tension Test 243
stressstrain curve like that of the structural steel, the area under the
curve can be approximated by either of the following equations:
Ut ~ GuCf (99)
or Ut ^ — 2 ^f (910)
For brittle materials the stressstrain curve is sometimes assumed to be
a parabola, and the area under the curve is given by
C/r~2,3V„e/ (911)
All these relations are only approximations to the area under the stress
strain curves. Further, the curves do not represent the true behavior in
the plastic range, since they are all based on the original area of the
specimen.
92. TruestressTruestrain Curve
The engineering stressstrain curve does not give a true indication of
the deformation characteristics of a metal because it is based entirely on
the original dimensions of the specimen, and these dimensions change
continuously during the test. Also, ductile metal which is pulled in
tension becomes unstable and necks down during the course of the test.
Because the crosssectional area of the specimen is decreasing rapidly at
this stage in the test, the load required to continue deformation falls off.
The average stress based on original area likewise decreases, and this
produces the fallofT in the stressstrain curve beyond the point of maxi
mum load. Actually, the metal continues to strainharden all the way
up to fracture, so that the stress required to produce further deformation
should also increase. If the true stress, based on the actual crosssectional
area of the specimen, is used, it is found that the stressstrain curve
increases continuously up to fracture. If the strain measurement is also
based on instantaneous measurements, the curve which is obtained is
known as a truestresstruestrain curve. This is also known as a flow
curve (Sec. 32) since it represents the basic plasticflow characteristics
of the material. Any point on the flow curve can be considered as the
yield stress for a metal strained in tension by the amount shown on the
curve. Thus, if the load is removed at this point and then reapplied, the
material will behave elastically throughout the entire range of reloading.
The true stress is the load at any instant divided by the crosssectional
area of the specimen at that instant.
0 = ^ (912)
244 Applications to Materials Testing
True strain was defined in Sec. 33 as
e = In ^^ = m — 7
Lo A
[Chap. 9
(913)
This definition of strain was proposed by Ludwik^ near the beginning of
140
130
120
110
100
90
'in
o80
o
o
 70
cn
tn
?i 60
■♦
50
40
30
20
10
.C^.
^py
.y
^
'A
^^
Corrected
necking
for
Max/'/
' loc
num
^^A
^
^^
'^^
/
.M
aximu
m lot
id
/^
^
{__
Sr
t^'nee
r/ng^^
'''ress
^^ro/n
Curt/
—
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.
Strain, in./in.
1.2
Fig. 93. Comparison of engineering and true stressstrain curves for nickel.
the century. It was also shown previously (Sec. 33) that the relation
between true strain and conventional linear strain is given by
e = In (e H 1)
(914)
The true stress may be determined from the average engineering stress
as follows:
Ai
F_A^
A^Ai
ip. Ludwik, "Elemente der technologischen Mechanik," SpringerVerlag OHG,
Berlin, 1909.
Sec. 92] The Tension Test 245
But, by the constancyof volume relationship,
From Eq. (914)
Ai Lo
e = In ^ = In (e + 1)
L
L Ao ..
or 7~="r = ^+l
cr = £ (e + 1) (915)
Figure 93 compares the true stressstrain curve for a nickel specimen
with its engineering stressstrain curve. Note that the large scale on the
strain axis, which was used to emphasize the plastic region, has com
pressed the elastic region into the y axis. Frequently, the true stress
strain curve is linear from the maximum load to fracture, while in other
cases its slope continuously decreases up to fracture. Little significance
should be attached to this linear region of the flow curve. When neck
ing occurs, the triaxial state of stress that is created in this region increases
the average longitudinal stress needed to continue plastic flow. There
fore, the shape of the flow curve from maximum load to fracture depends
on the rate of development of the neck. This can be different for mate
rials with different strainhardening behavior, and therefore there is no
assurance that the flow curve will be linear in this region.
The following parameters are usually determined from the true stress
strain curve.
True Stress at Maximum Load
The true stress at maximum load corresponds to the true tensile
strength. For most materials necking begins at maximum load. As a
good approximation, necking will occur at a value of strain where the
true stress equals the slope of the flow curve. Let am and e„ denote the
true stress and strain at maximum load, while Am represents the cross
sectional area of the specimen at maximum load. Then,
P P An
■* max ^ max i ^^0
O^u — — j Cm — —A iu — m r—
Ao /im Am
and o„ = (Tm exp (€„) (916)
Equation (916) relates the ultimate tensile strength to the true stress
and strain at maximum load.
246 Applications to Materials Testing [Chap. 9
True Fracture Stress
The true fracture stress is the load at fracture divided by the cross
sectional area at fracture. This stress should be corrected for the tri
axial state of stress existing in the tensile specimen at fracture. Since
the data required for this correction are often not available, truefracture
stress values are frequently in error.
True Fracture Strain
The true fracture strain e/ is the true strain based on the original area
Ao and the area after fracture, Af.
e/ = ln^^ (917)
This parameter represents the maximum true strain that the material
can withstand before fracture and is analogous to the total strain to
fracture of the engineering stressstrain curve. .Since Eq. (914) is not
valid beyond the onset of necking, it is not possible to calculate e/ from
measured values of e/. However, for cylindrical tensile specimens the
reduction of area, q, is related to the true fracture strain by the
relationship
5=1 exp (e/) (918)
True Uniform Strain
The true uniform strain e„ is the true strain based only on the strain
up to maximum load. It may be calculated from either the specimen
crosssectional area Au or the gage length L„ at maximum load. Equa
tion (914) may be used to convert conventional uniform strain to true
uniform strain. The uniform strain is often useful in estimating the
formability of metals from the results of a tension test.
e„ = In ^ (919)
True Local Necking Strain
The local necking strain e„ is the strain required to deform the speci
men from maximum load to fracture.
en = In 4^ . (920)
The usual method of determining a true stressstrain curve is to meas
Sec. 92] The Tension Test 247
ure the crosssectional area of the specimen at increments of load up to
fracture. Micrometers or special dial gages can be used. Care should
be taken to measure the minimum diameter of the specimen. This
method is applicable over the complete range of "^treag to fracture,
including the region after necking has occurred. However, lb correct
precisely for the complex stresses at the neck, it is necessary to know
the profile of the contour of the neck. This method of determination is
limited to fairly slow rates of strain and to tests at room temperature.
If the specimen has a circular cross section, the true strain may be
readily calculated from the original diameter Do and the instantaheous
diameter A.
' = '4. = '^T = '"t^ = 21„g! (921).
True stress and strain may also be determined from the conventional
stress and strain by means of Eqs. (914) and (915). The use of these
equations implies that the axial strain is uniformly distributed over the
gage length of the specimen, since their derivation is based on the con
stancyofvolume relationship. Stresses and strains determined from
these equations are accurate up to the beginning of necking, but beyond
this stage the major portion of the strain is localized at the neck and the
equations do not apply.
True stressstrain curves may be obtained at high strain rates and at
elevated temperatures by using the twoload method.^ Diameters at
various positions along tapered specimens are measured before and after
testing. The true stress acting at each location on the specimen is the
maximum load divided by the area at that point after testing. This
gives the flow curve from the state of yielding to the point of maximum
load. If the load at fracture is measured, the curve can be extended to
the fracture stress by linear extrapolation.
It is usually desirable to be able to express the true stressstrain curve
by a mathematical relationship. The simplest useful expression is the
power curve described earlier in Sec. 32.
(7 = Xe" (922)
where n is the strainhardening coefficient and K is the strength coefficient.
A loglog plot of true stress and strain up to maximum load will give a
straight line if this equation is satisfied by the data (Fig. 94). The
linear slope of this line is n, and K is the true stress at e = 1.0. In
order to make the data fall closer to a straight line, it is usually desirable
1 C. W. MacGregor, J. Appl. Mech., vol. 6, pp. A156158, 1939.
248 Applications to Materials Tcsti.
[Ch
ap.
."T
K
to subtract the elastic strain from the
total strain. Some typical values of n
and K are listed in Table 93.
There is nothing basic about Eq.
(922), so that frequent deviations from
this relationship are observed. One
common type of deviation is for a
loglog plot of Eq. (922) to result in
two straight lines with different slopes,
while in other cases a curve with con
tinuously changing slope is obtained.
Equation (923) is typical of the more
complicated relationships which have been suggested^ to provide better
agreement with the data.
0.001
0.01 0.1
True strain f
Fig. 94. loglog plot of true stress
strain curve.
= Ci  (Ci  (72) exp 
(A)
(923)
Table 93
Values of n and K for Metals at Room Temperature
Metal
Condition
n
K, psi
Ref.
0.05% C steel
SAE 4340 steel
0.6% C steel
0.6% C steel
Copper
70/30 brass
Annealed
Annealed
Quenched and tempered 1000°F
Quenched and tempered 1300°F
Annealed
Annealed
0.26
0.15
0.10
0.19
0.54
0.49
77,000
93,000
228,000
178,000
46,400
130,000
t J. R. Low and F. Garofalo, Froc. Soc. Exptl. Stress Anal., vol. 4, no. 2, pp. 1625,
1947.
t J. R. Low, "Properties of Metals in Materials Engineering," American Society
for Metals, Metals Park, Ohio, 1949.
93. Instability in Tension
Necking generally begins at maximum load during the tensile defor
mation of a ductile metal. ^ An ideal plastic material in which no strain
hardening occurs would become unstable in tension and begin to neck
just as soon as yielding took place. However, a real metal undergoes
1 E. Voce, Metallurgia, vol. 51, pp. 219226, 1955.
2 An exception to this is the behavior of coldrolled zirconium tested at 200 to 370°C,
where necking occurred at a strain of twice the strain at maximum load. See J. H.
Keeler, Trans. ASM, vol. 47, pp. 157192, 1955, and discussion by A. J. Opinsky,
pp. 189190.
Sec. 93]
TheT
ension
Test
249
strain hardening, which tends to increase the loadcarrying capacity of
the specimen as deformation increases. This effect is opposed by the
gradual decrease in the crosssectional area of the specimen as it elon
gates. Necking or localized deformation begins at maximum load, where
the increase in stress due to decrease in the crosssectional area of the
specimen becomes greater than the increase in the loadcarrying ability
b
^
in
^
1
a>
^
f
/
u /
CTm
^^ i
1
t/^
y
+ 1
" 1
/I
B
^ —
Conventiona
Cu
strain e >
<
1 + e,, 
>■
Fig. 95. Considere's construction for the determination of the point of maximum load.
of the metal due to strain hardening. This condition of instability lead
ing to localized deformation is defined by the condition dP = 0.
P = aA
dP = a dA \ A da =
From the constancyofvolume relationship,
Therefore
dL dA
L A
dA dL da J
A L a
de
1 +e
da
Te='
(924)
da a
(925)
de 1 + e
or
Equation (924) says that necking will occur in uniaxial tension at a strain
at which the slope of the true stressstrain curve equals the true stress at
that strain.
Equation (925) permits an interesting geometrical construction for
the determination of the point of maximum load.^ In Fig. 95 the stress
' A. Considere, Ann. ponts et chauss^es, vol. 9, ser. 6, pp. 574775, 1885.
250 Applications to Materials Testing [Chap. 9
strain curve is plotted in terms of true stress against conventional linear
strain. Let point A represent a negative strain of 1.0. A line drawn
from point .4 which is tangent to the stressstrain curve will establish
the point of maximum load, for according to Eq. (925) the slope at this
point is a /{I + e). The stress at this point is the true stress at maxi
mum load, (T,„. If we had plotted average stress, this would have been
the tensile strength a„. The relation between these two stresses is
a„ _ A _ Lq
(Tm Ao L
From the definition of conventional linear strain,
Lo 1
L 1 +e
so that a„ = am :7— 7 — (926)
A study of the similar triangles in Fig. 95 shows that Eq. (926) is satis
fied when OD is the tensile strength.
If the flow curve for a material is given by the power law of Eq. (922),
it is possible readily to determine the strain at which necking occurs.
a = /ve"
^ = 0 = Ae" = nKe^~^
de
6„ = n (927)
Therefore, the strain at which necking occurs is numerically equal to the
strainhardening coefficient.
Plastic instability is often important in forming operations with sheet
metal since the strain at which the deformation becomes localized con
stitutes the forming limit of the metal. Lankford and SaibeP have deter
mined the criteria for localized deformation for the case of a sheet sub
jected to biaxial tensile forces (stretching), a thin wall tube subjected to
internal pressure and axial loading, and a sheet subjected to a hydrostatic
bulge test.
94. Stress Distribution at the Neck
The formation of a neck in the tensile specimen introduces a complex
triaxial state of stress in that region. The necked region is in effect a
mild notch. As was discussed in Sec. 712, a notch under tension pro
1 W. T. Lankford and E. Saibel, Trans. AIME, vol. 171, pp. 562573, 1947.
)ec,
94]
The Tension Test
251
o>
*ar
duces radial and transverse stresses which raise the value of longitudinal
stress required to cause plastic flow. Therefore, the average true stress
at the neck, which is determined by dividing the axial tensile load by the
minimum crosssectional area of
the specimen at the neck, is higher
than the stress which would be
required to cause flow if simple
tension prevailed. Figure 96 illus
trates the geometry at the necked
region and the stresses developed
by this localized deformation. R
is the radius of curvature of the
neck, which can be measured either
by projecting the contour of the
necked region on a screen or by
using a tapered, conical radius
gage.
Bridgman' made a mathematical
analysis which provides a correction
to the average axial stress to compensate for the introduction of trans
verse stresses. This analysis was based on the following assumptions :
1 . The contour of the neck is approximated by the arc of a circle.
2. The cross section of the necked region remains circular throughout
the test.
3. The Von Mises criterion for yielding applies.
4. The strains are constant over the cross section of the neck.
<^t
Fig. 96. (a) Geometry of necked region;
(6) stresses acting on element at point 0.
Table 94
Correction Factors to Be Applied to Average True Stress to
Compensate for Transverse Stresses at Neck of
Tensile Specimen
a/R
Bridgman
Davidenkov
factor
factor
1.000
1.000
Vz
0.927
0.923
M
0.897
0.889
1
0.823
0.800
2
0.722
0.667
3
0.656
0.571
4
0.606
0.500
' p. W. Bridgman, Trans. ASM, vol. 32, p. 553, 1944.
252 Applications to Materials Testing
[Chap. 9
According to this analysis the ratio of the true axial stress o to the
average axial stress oav is
(7av (1 + 2R/a)\ln (1 + a/2R)]
(928)
Davidenkov and Spiridonova^ determined a correction for necking based
on somewhat different assumptions
from those of Bridgman. Their
expression is given by
a
1
1 + a/4R
(929)
These two equations differ by less
than 1 per cent for values of a/R
less than 0.6. Tj^pical values for
these corrections are given in
Table 94.
The determination of the radius
of curvature of the neck during the
progress of the test is certainly
not a routine or easy operation.
In order to help with this situa
tion, Bridgman determined an em
pirical relationship between the
neck contour (a/R) and the true
strain, based on about 50 steel
specimens. Figure 97 shows this
relationship converted into the var
iation of a/a^y with true strain.
Experimental values^ for copper and steel are also included in this figure.
This investigation showed that Bridgman's equation provides better
agreement with experiment than Davidenkov's. The dashed curve in
Fig. 93 is the true stressstrain curve of nickel adjusted for necking
by means of Bridgman's correction factor. The problem of the stress
distribution at the neck of flat tensile specimens has been considered by
Aronofsky.^
Fig. 97. Relationship between Bridgman
correction factor a/aav and true tensile
strain. (E. R. Marshall and M. C. Shaw,
Trans. ASM, vol. 44, p. 716, 1952.)
95. Strain Distribution in the Tensile Specimen
The strain distribution along the length of a tensile specimen is not
uniform, particularly in metals which show pronounced necking before
1 N. N. Davidenkov and N. I. Spiridonova, Proc. ASTM, vol. 46, p. 1147, 1946.
2 E. R. Marshall and M. C. Shaw, Trans. ASM, vol. 44, pp. 705725, 1952.
3 J. Aronofsky, J. Appl. Mech., vol. 18, pp. 7584, 1951.
Sec. 95]
fheT
ension
Test
253
fracture. Figure 98 shows in a schematic way the distribution of local
elongation along the length of a tensile specimen. The exact distribu
tion of strain will depend upon the
metal, the gage length, and the
shape of the cross section of the
test section. In general, the softer
and more ductile the metal, the
greater the amount of deformation
away from the necked region.
Also, the shorter the gage length,
the greater the influence of local
ized deformation at the neck on the
total elongation of the gage length.
Therefore, for a given material, the
shorter the gage length, the greater
the percentage elongation. It is
for this reason that the specimen gage length should always be reported
with the percentage elongation.
It is generally recognized that in order to compare elongation measure
ments of differentsized specimens, the specimens must be geometrically
similar; i.e., the ratio of gage length to diameter must be constant. In
the United States the standard tensile specimen has a 0.505in. diameter
and a 2in. gage length. Thus, L/D ~ 4, or L = 4.51 y/A. This is the
basis for the dimensions of the ASTM tensile specimens listed in Table
95. The British standards specify L/D = 3.54, while the German
standards use L/D = 10.
Table 95
Dimensions of ASTM Tensile Specimens
Gage length
Fig. 98. Schematic drawing of variation
of local elongation with position along
gage length of tensile specimen.
Diam, in.
Gage
length, in.
L/D
0.505
2
3.97
0.357
1.4
3.92
0.252
1
3.97
0.160
0.634
3.96
For tensile specimens cut from thin sheet the ratio of width to thick
ness can affect the total elongation. With a constant gage length, an
increase in either the width or the thickness of the specimen results in an
increase in elongation. However, so long as the width and thickness
are varied without changing the crosssectional area, the elongation is
not affected. Available data indicate that the percentage elongation
increases in proportion to the area raised to a fractional power.
254 Applications to Materials Testing
[Chap. 9
The uniform elongation is not affected by specimen geometry, since
up to maximum load the specimen elongates and contracts in diameter
uniformly. The specimen changes from a cylinder of a certain length
and diameter to a cylinder of longer length and smaller diameter. For
this reason the uniform elongation is a more fundamental measure of
ductility than the conventional percentage elongation.
96. Effect of Strain Rate on Tensile Properties
The roomtemperature stressstrain curve is not greatly influenced by
changes in the rate of straining of the order obtainable in the ordinary
■^40
o
o
o
sz
a>
c=
^ 10
n
Ro
om fe
vpera
fure
^
^
^
"^
_200^C^
/
40C
^^q'^
bC
/
8C
lf^>i
10'
10"
^ 102
Strain rate, sec
10'
Fig. 99. Effect of strain rate on the tensile strength of copper for tests at various tem
peratures. {A. Nadai and M. J. Manjoine, J. Appl. Mech., vol. 8, p. A82, 1941.)
tp.nsion test. (The effect of impact and very highspeed loading will be
T'nsidered in Chap. 14.) Highspeed tensile tests^~^ in which the rate
of loading has been varied by a factor of about 100,000 have shown that
the yield stress is more sensitive to increases in strain rate than the tensile
strength. High rates of strain cause a yield point to appear in specimens
of lowcarbon steel which do not show a yield point under ordinary rates
of loading. The effect of strain rate in raising resistance to deformation
,j;enerally increases on testing at elevated temperature. Figure 99 shows
the effect of strain rate on the tensile strength of copper at various
temperatures.
The determination of a mathematical relationship between the flow
■ J. Winlock, Trans. AIME, vol. 197, pp. 797803, 1953.
' R. J. MacDonald, R. L. Carlson, and W. T. Lankford, Proc. ASTM, vol. 56, pp.
704723, 1956.
« A. Nadai and M. J. Majoine, J. Appl. Mech., vol. 8, pp. A77A91, 1941.
>ec,
96] The Tension Test 255
stress and the strain rate is difficult because of the many experimental
problems associated with measuring tensile properties at very rapid rates
of deformation. Among the experimental problems is that an adiabatic
condition is created at high strain rates, causing the temperature of the
specimen to increase; there is not enough time for the heat of plastic
deformation to be dissipated. Tests in which the specimen is pulled at
a constant true strain rate are not readily performed on conventional
testing machines. Although it is fairly easy to maintain a constant rate
of crosshead movement, this does not ensure a constant rate of strain in
the specimen since the rate of straining in the specimen increases with
load, particularly during necking.
Nadai^ has presented a mathematical analysis of the conditions exist
ing during the extension of a cylindrical specimen with one end fixed and
the other attached to the movable crosshead of the testing machine. The
crosshead velocity is y = dL/dt. The strain rate expressed in terms of
conventional linear strain is e.
^ dt dt U dt U ^^'^^^
Thus, the conventional strain rate is proportional to the crosshead veloc
ity. The equation is applicable up to the onset of necking.
The true strain rate e is given by
^e d[\n (L/Lo)] 1 dL v
dt dt L dt L
(931)
This equation indicates that for a constant crosshead speed the true
strain rate will decrease as the specimen elongates. To maintain a con
stant true strain rate, the crosshead velocity must increase in proportion
to the increase in length of the specimen. For a cylindrical specimen
the true strain rate is related to the instantaneous diameter Di by
de d[2 In (Z)o/A)] 2 d{D^)
dt dt Di dt
(932)
The true strain rate is related to the conventional strain rate by the
following equation :
. _ v^ _ Lode _ 1 de _ e
'~L~Ldt~l+edt~l+e ^^"'^'^^
Strainrate experiments with mild steel have shown a semilogarithmic
relationship between the lower yield point and the strain rate.
(To = ki + ki log e (934)
' A. Nadai, "Theory of Flow and Fracture of Solids," vol. I, pp. 7475, McGraw
Hill Book Company, Inc., New York, 1950.
256
Applications to Materials Testing
[Chap. 9
However, a more general relationship ^ between flow stress and strain
rate, at constant temperature and strain, seems to be
C{ey
W,T
(935)
where m is a coefficient known as the strainrate sensitivity. The strain
rate sensitivity m may be defined as the ratio of the incremental change in
log <T to the resultant change in log e, at a given strain and temperature.
A value for this parameter can be obtained from a test where the strain
rate is rapidly changed from one value to another.
m —
log (q2/oi)
log (€2/61)
(936)
The strainrate sensitivity for most metals increases with temperature
and with strain.
97. Effect of Temperature on Tensile Properties
In general, strength decreases and ductility increases as the test tem
perature is increased. However, structural changes such as precipitation,
/96°C
Strain e
Fig. 910. Changes in engineering stress
strain curves of mild steel with tempera
ture.
en f
• O
O) o
'Room temperature
Tens,
Temperature — *
Fig. 91 1 . Variation of tensile properties
of steel with temperature.
strain aging, or recrystallization may occur over certain temperature
ranges to alter this general behavior. Further, extended exposure at
elevated temperature may cause creep.
The change with temperature in the shape of the engineering stress
strain curve in mild steel is shown schematically in Fig. 910. The vari
ation of the tensile properties of steel with temperature is shown in Fig.
911. The strength of steel increases as the temperature is raised above
room temperature. The maximum in strength is accompanied by a mini
1 C. Zener and J. H. Hollomon, /. App/. Phy&., vol. 15, pp. 2232, 1944.
Sec. 97]
TheL
Test
257
mum in ductility, which occurs in the vicinity of 400°F, due to strain
aging or blue brittleness. Figure 912 shows the variation of yield
strength with temperature for bodycentered cubic tantalum/ tungsten,
molybdenum, and iron and facecentered cubic nickel. Note that the
yield strength of nickel increases with decreasing temperature to a lesser
extent than in the bodycentered cubic metals. This difference in the
140
120
S.100
o
o
Sso
£ 60
40
20=
200
\
\
\
1
\
\
\\"
\
\
\
f"^
K
\
Mi
\
\
\
\
:;^^.
.^
>
^*=
—
200 400
Temperoture, °C
600
800
Fig. 91 2. Effect of temperature on the yield strength of bodycentered cubic Ta, W,
Mo, Fe and facecentered cubic Ni. (/. H. Bechtold, Acta Met., vol. 3, p. 252, 1955.)
temperature dependence of yield strength is believed to be of significance
in explaining why facecentered cubic metals do not exhibit brittle frac
ture at low temperatures. The horizontal portion of the curves for W
and Mo at low temperature represents the brittlefracture strength, for
these metals undergo brittle fracture without extensive yielding at these
temperatures. In comparing the flow stress or yield stress of a material
at two temperatures it is advisable to correct for the effect of temperature
on elastic modulus by comparing ratios of a/E rather than simple ratios
of flow stress.
Figure 913 shows the variation of the reduction of area with temper
ature for these same metals. Note that tungsten is almost completely
brittle at 200°C, iron at — 200°C, while nickel decreases little in ductility
over the entire temperature range. The lack of a brittle transition in
nickel is a general characteristic of facecentered cubic metals and corre
lates with its small temperature dependence of yield strength. The
behavior of bodycentered tantalum is anomalous in this respect, for it
shows no ductility transition although the yield stress increases rapidly
at low temperature.
1 J. H. Bechtold, Acta Met., vol. 3, pp. 249254, 1955.
258 Applications to Materials Testing
[Chap.:9
The temperature dependence of the flow stress at constant strain and
strain rate can be generally represented by
a = C2. exp
RT
(937)
where Q = an activation energy for plastic flow, cal/g mole
R = universal gas constant, 1.987 cal/(deg)(mole)
T = testing temperature, °K
If this equation represents the data, a straight line is obtained for a plot
of In a versus 1/T. The activation energy is obtained from the slope of
100
60
40
20
j^
—
7^'
Jc
]
^
V
1
i
r
w/
/
J
'Mo
/
/
\
J
V
200
200 400
Temperature, °C
600
800
Fig. 91 3. Effect of temperature on the reduction of area of Ta, W, Mo, Fe, and Ni.
(/. H. Bechtold, Acta Met, vol. 3, p. 253, 1955.)
the line. Equation (937) has been found valid for steel, molybdenum,
and tungsten over a considerable temperature range. It is, however, not
valid ^ at temperatures below around 100°K.
98. Combined EFFect of Strain Rate and Temperature
Zener and Hollomon^ suggested that the flow stress at constant strain
was related to both the strain rate and the temperature in the following
way:
(e exp If)
(938)
AH is an activation energy expressed in calories per gram mole and is
related to the activation energy Q in Eq. (937) by Q = m AH, where
1 E. T. Wessel, Trans. ASM, vol. 49, pp. 149172, 1957.
^ Zener and Hollomon, op. cit.
Sec. 98] The Tension Test 259
m is the strainrate sensitivity. The quantity in parentheses in Eq.
(938) is often called the ZenerHollomon parameter Z.
A/7
Z = e exp ^ (939)
A plot of In e versus 1/T should result in a straight line. Zener and
Hollomon originally based this relationship on the fact that the yield
strength and tensile strength of steel and copper correlated well with Z
over rather wide ranges of e and T. More recently it has been found to
hold for true stress data on molybdenum^ and pure aluminum. It has
been shown that the same functional relationship is obtained between
stress and strain for a constant value of Z, but since AH is not inde
pendent of stress this relationship does not uniquely describe the flow
curve.
A slightly different type of approach to this problem was taken by
MacGregor and Fisher.^ They proposed that strain rate and temper
ature could be combined into a velocitymodified temperature. Then, the
flow stress at a particular strain is a function of the velocitymodified
temperature T^.
where T, = t(\  k In  ) (940)
In Eq. (940) k and eo are constants related to reactionrate constants.
This equation was originally verified for data on steel and aluminum
over a large temperature range but only a small range of strain rates.
More recently* it has been verified for lowcarbon steel over a greater
range of strain rates.
When Eq. (938) was first proposed, it w^as interpreted much more
broadly than it is today. It was suggested that Eq. (938) represented
a mechanical equation of state, analogous to the equations of state for a
perfect gas. The concept of the mechanical equation of state^ indicated
that the flow stress of a metal was a function only of the instantaneous
values of strain, strain rate, and temperature regardless of the previous
temperature and rate of straining. In other words, if a metal did not
undergo a phase change or gross change in metallurgical structure, it was
considered that a metal would arrive at the same final conditions of flow
• J. H. Bechtold, Trans. AIME, vol. 197, pp. 14691475, 1953.
2 T. A. Trozera, O. D. Sherby, and J. E. Dorn, Trans. ASM, vol. 49, pp. 173188,
1957.
3 C. W. MacGregor and J. C. Fisher, /. Appl. Mech., vol. 13, pp. 1116, 1946.
^ MacDonald, Carlson, and Lankford, op. cit.
U. H. Hollomon, Trans. AIME, vol. 171, p. 535, 1947.
260
Applications to Materials Testing
[Chap. 9
stress and strain by dilYerent paths of strain rate and temperature pro
vided that Eq. (93S) was satistied. However, extensive experiments' 
on aluminum, copper, stainless steel, and lowcarbon steel have shown
appreciable deviations from the beha^•ior predicted by the mechanical
equation of state. It is now established that the flow stress depends on
the previous conditions of temperature and strain rate as well as on the
instantaneous values of strain, strain rate, and temperature. The failure
of the mechanical equation of state is due to the fact that the structural
changes occurring during plastic deformation are dependent not solely on
strain but on the strain rate and temperature as well.
99. Notch Tensile Test
The ordinary tension test on smooth specimens will fail to indicate
notch sensitivity in metals. However, a tension test with a notched
P^
a
D ^
^ P
Fig. 914. Details of notched tensile specimen.
tensile specimen will show whether or not a material is notchsensitive
and prone to brittle fracture in the presence of a stress concentration.
Notch sensitivity can also be investigated b^y means of the notched
impact test, as described in Chap. 14; this test has been widely used for
mild steels and as a test for temper embrittlement. The impact test has
the advantage of ease in preparing specimens and testing over a wide
range of temperature, but it lacks the advantage of the notch tensile test
of more basic interpretation of test results because of a betterdefined
state of stress. The notch tensile test has been used for testing high
strength steels, for studying hydrogen embrittlement in steels, and for
investigating the notch sensitivit}' of hightemperature alloys.
Figure 914 shows the geometric details of a notched tensile specimen.
The introduction of the notch produces a condition of biaxial stress at
the root of the notch and triaxial stress at the interior of the specimen.
As was shown previously in Sec. 712, the presence of transverse stress
at the notch increases the resistance to flow and decreases the ratio of
» J. E. Dorn, A. Goldberg, and T. E. Tietz, Trans. AIME, vol. 180, p. 205, 1949.
 T. E. Tietz and J. E. Dorn, "Cold Working of Metals," pp. 163179, American
Society for Metals, Metals Park, Ohio, 1949.
Sec. 99]
The Tension Test
261
shear stress to tensile stress. A notch is characterized by the notch
sharpness a/r and the notch depth.
Notch depth = 1  r,2
200
100
The notch strength is defined as the maximum load divided by the original
crosssectional area at the notch. The notchstrength ratio (NSR) is the
ratio of the notch strength to the
ultimate tensile strength. The '^^^
NSR is a measure of notch sensi
tivity. If the NSR is less than S.
unity, the material is notchbrittle. ^
The term notch ductility is used
to indicate the reduction of area
at the notched region . The amount
of notch ductility is often very
small and therefore is difficult to
determine accurately. The most
commonly used notch has 50 per
cent of the area removed at the
notch, with a radius of 0.001 in.
and a 60° notch angle.
The notch sensitivity of steel is
usually evaluated by measuring
the notch strength as a function
of tensile strength. Figure 915
shows the type of curves which are
obtained. The notch strength
drops off sharply at about the
200,000psi strength level, indi
cating that the steels are notch
brittle above this strength level. Below this point the NSR is about
1.5. Note that the notch ductility decreases to very low values for tensile
strengths over 200,000 psi. For most heattreated steels the NSR falls
off below 1.5 when the notch ductility drops below about 6 per cent.
The notchstrengthtensilestrength curve is a function of the notch
shape. Increasing the notch radius reduces the elasticstress concentra
tion but has little effect on the degree of triaxiality of stress. The effect
of changes in notch radius on notched tensile properties depends on the
strength level of the steel. At high strength levels, where ductility is low,
reducing the notch sharpness increases the notch strength and the notch
strength ratio. At strength levels below about 200,000 psi there is no
effect on the notch strength of increasing the radius from 0.001 to 0.050 in.
/■
— ,
V
/
^
\
V
/
\
/
\
,4
/
/
/
\Y
X\
\
\
V
^
100 200
Tensile strength, 1,000 psi
300
Fig. 91 5. Notch tensile properties of two
steels. Steel ^4 has higher notch sensi
tivity than steel B.
262 Applications to Materials Testing [Chap. 9
On the other hand, changing the notch depth produces large changes in
triaxiality with only small changes in stress concentration. At low
strength levels the notchstrength ratio is a linear function of the notch
depth.
notch depth, %
NSR = 1 +
100
At higher strengths, where the ductility is low, the notch strength is
dependent on the notch ductility. The literature on notch tensile testing
has been described in a number of reviews. ^'^
910. Tensile Properties of Steels
Because of the commercial importance of ferrous materials, a great deal
of work has been done in correlating their tensile properties with composi
tion and microstructure. It has been clearly demonstrated that micro
structure is the chief metallurgical variable which controls the tensile
properties of steels. Because of the wide variety of microstructures which
are possible with changes in composition and heat treatment, this is a very
interesting, yet somewhat complex, subject.
The tensile properties of annealed and normalized steels are controlled
by the flow and fracture characteristics of the ferrite and by the amount,
shape, and distribution of the cementite. The strength of the ferrite
depends on the amount of alloying elements in solid solution (see Fig. 59)
and the ferrite grain size. The carbon content has a very strong effect
because it controls the amount of cementite present either as pearlite or
as spheroidite. The strength increases and ductility decreases with
increasing carbon content because of the increased amount of cementite
in the microstructure. A normalized steel will have higher strength than
an annealed steel because the more rapid rate of cooling used in the
normalizing treatment causes the transformation to pearlite to occur at a
lower temperature, and a finer pearlite spacing results. Differences in
tensile properties due to the shape of the cementite particles are shown in
Fig. 916, where the tensile properties of a spheroidized structure are
compared with a pearlitic structure for a steel with the same carbon
content. Empirical correlations have been worked out^ between composi
tion and cooling rate for predicting the tensile properties of steels with
pearlitic structures.
ij. D. Lubahn, Notch Tensile Testing, "Fracturing of Metals," pp. 90132,
American Society for Metals, Metals Park, Ohio, 1948.
2 J. D. Lubahn, Trans. ASME, vol. 79, pp. 111115, 1957.
3 1. R. Kramer, P. D. Gorsuch, and D. L. Newhouse, Trans. AIMS, vol. 172,
pp. 244272, 1947.
Sec. 910]
TheT
ension
Test
263
200
•^ 'J^
"^
^
^
p'i
> "^
y'
''^
'^
hn^^:
v^/
V
\
o 150
o
o
/
,^^'^
U^ ..nth .
' —
Lamellar
1 1
\^
A
1^ 
"^ ^
Reducting
of area \
Stress, 1
5
o
o'<a
^
IV'
.■'
^■~~
 —
•
^
50
/
^
.yiel
dst^j
>
^
\
/
,'
^
1
Elongation
^ 1
^''
'
^^
::r::
20
25
30 35 20 25
Rockwell hardness, C scale
30
50
40
30
20
10
35
Fig. 916. Tensile properties of pearlite and spheroidite in a eutectoid steel. {From
E. C. Bain, "Alloying Elements in Steel," p. 39, American Society for Metals, Metals
Park, Ohio, 1939.)
One of the best ways to increase the strength of annealed steel is by
cold working. Table 96 gives the tensile properties which result from
the cold reduction of SAE 1016 steel bars by drawing through a die.
Table 96
Effect of Cold Drawing on Tensile Properties of
SAE 1016 STEELf
Reduction
Yield
Tensi'.e
Elongation,
Reduction
of area by
strength.
strength.
in 2 in.,
of area,
drawing, %
psi
psi
%
%
40,000
66,000
34
70
10
72,000
75,000
20
65
20
82,000
84,000
17
63
40
86,000
95,000
16
60
60
88,000
102,000
14
54
80
96,000
115,000
7
26
t L. J. Ebert, "A Handbook on the Properties of Cold Worked Steels," PB 121662,
Office of Technical Services, U.S. Department of Commerce, 1955.
The pearlitic structure in steel can be controlled best by transforming
the austenite to pearlite at a constant temperature instead of allowing it
to form over a range of temperature on continuous cooling from above
the critical temperature. Although isothermal transformation is not in
widespread commercial use, it is a good way of isolating the effect of
264 Applications to Materials Testing
[Chap. 9
certain microstructures on the properties of steel. Figure 917 shows the
variation of tensile properties for a NiCrMo eutectoid steel with iso
thermalreaction temperature.^ This is a recent extension of Gensamer's
work," which showed that the tensile strength varied linearly with the
logarithm of the mean free ferrite path in isothermally transformed struc
tures. In the region 1300 to 1000°F the transformation product is
300
8 250
o
■&200
^ 150
100
50

>
/ /
4
^%,
A^~\'
/
x^
^ \.
1 I
1 1
J^longarfon
I I I 1 1
60
40,
20
1300 1100 900 700
Transformation temperature, "F
500
Fig. 91 7. Relationship of tensile properties of NiCrMo steel to isothermaltrans
formation temperature. {E. S. Davenport, Trans. AIME, vol. 209, p. 684, 1957.)
lamellar pearlite. The spacing between cementite platelets decreases
with transformation temperature and correspondingly the strength
increases. In the region 800 to 500°F the structure obtained on trans
formation is acicular bainite. The bainitic structure becomes finer with
decreasing temperature, and the strength increases almost linearly to
quite high values. Good ductility accompanies this high strength over
part of the bainite temperature range. This is the temperature region
used in the commercial heattreating process known as austempering.
The temperature region 1000 to 800°F is one in which mixed lamellar
and acicular structures are obtained. There is a definite ductility mini
mum and a leveling off of strength for these structures. The sensitivity
1 E. S. Davenport, Trans. AIME, vol. 209, pp. 677688, 1957.
2 M. Gensamer, E. B. Pearsall, W. S. Pellini, and J. R. Low, Trans. ASM, vol. 30,
pp. 9831020, 1942.
Sec. 910]
TheT
ension
Test
265
of the reduction of area to changes in microstructiire is well illustrated
by these results.
The best combination of strength and ductility is obtained in steel
which has been quenched to a fully martensitic structure and then
tempered. The best criterion for comparing the tensile properties of
quenched and tempered steels is on the basis of an asquenched struc
ture of 100 per cent martensite. However, the attainment of a com
pletely martensitic structure may, in many cases, be commercially
/I 99.9% martensite _
95 % martensite
C  90% martensite
D  80% martensite _
E  50 % martensite
0.1
0.2
0.3 0.4 0.5 0.6
Carbon content, wt. %
0.7
0.8
Fig. 918. Asquenched hardness of steel as a function of carbon content for different
percentages of martensite in the microstructure. {ASM Metals Handbook, 1948
ed., p. 497.)
impractical. Because of the importance of obtaining a fully marten
sitic structure, it is desirable that the steel have adequate hardenability.
Hardenahility, the property of a steel which determines the depth and
distribution of hardness induced by quenching, should be differentiated
from hardness, which is the property of a material which represents its
resistance to indentation or deformation. (This subject is discussed in
Chap. 11.) Hardness is associated with strength, while hardenability
is connected with the transformation characteristics of a steel. Harden
ability may be increased by altering the transformation kinetics by the
addition of alloying elements, while the hardness of a steel with given
transformation kinetics is controlled primarily by the carbon content.
Figure 918 shows the hardness of martensite as a function of carbon
content for different total amounts of martensite in the microstructure.
These curves can be used to determine whether or not complete harden
ing was obtained after quenching. Hardness is used as a convenient
measure of the strength of quenched and tempered steels. The validity
of this procedure is based on the excellent correlation which exists between
266 Applications to Materials Testing
[Chap. 9
tensile strength and hardness for heattreated, annealed, and normalized
steels (Fig. 919).
The mechanical properties of a quenched and tempered steel may be
altered by changing the tempering temperature. Figure 920 shows how
hardness and the tensile properties vary with tempering temperature for
an SAE 4340 steel. This is the typical behavior for heattreated steel.
Rockwell C
hardness
260
240
12 25 31 38 43
47 5
/•
/
220
/
f
/
/
.200
O)
o.
/
/
8180
Q.
/
/
/
/
strength,
o o
/ /
/
/
A
' /
••^ 120
/
/
/ /
C
/ /
^ 100
80
/
^
r
60
40
100
200 300 400
Brinell hardness number
500
Fig. 919. Relationship between tensile strength and hardness for quenched and
tempered, annealed, and normalized steels. (SAE Handbook.)
Several methods for correlating and predicting the hardness change in
different steels with tempering temperature have been proposed.^"* In
using tempering diagrams like Fig. 920, it is important to know whether
or not the data were obtained on specimens quenched to essentially 100 per
cent martensite throughout the entire cross section of the specimen.
Because of the variability in hardenability from heat to heat, there is no
assurance of reproducibility of data unless this condition is fulfilled.
A great many lowalloy steels have been developed and are used in the
quenched and tempered condition. A study of the tensile properties of
these steels could lead to considerable confusion were it not for the fact
1 J. H. Hollomon and L. D. Jaffe, Trans. AIME, vol. 162, p. 223, 1945.
2 R. A. Grange and R. W. Baughman, Trans. ASM, vol. 48, pp. 165197, 1956.
3 L. D. Jaffe and E. Gordon, Trans. ASM, vol. 49, pp. 359371, 1957.
Sec. 910]
TheT«
Test
267
that certain generalities can be made about their properties.^ For low
altoy steels containing 0.3 to 0.5 per cent carbon which are quenched to
essentially 100 per cent martensite and then tempered back to any given
tensile strength in the range 100,000 to 200^000 psi, the other common
tensile properties will have a relatively fixed value depending only on
the tensile strength. In other words, the mechanical properties of this
300
250
'200
150
^100
50
E\o^
ion
60.
50
40.5
30.
20
 10uj
200 400 600 800 1000 1200 1400
Tempering temperature, °F
Fig. 920. Tensile properties of quenched and tempered SAE 4340 steel as a function of
tempering temperature. For fully hardened lin.diameter bars.
important class of steels do not depend basically on alloy content, carbon
content within the above limits, or tempering temperature. It is impor
tant to note that this generalization does not say that two alloy steels
given the same tempering treatment will have the same tensile properties,
because different tempering temperatures would quite likely be required
to bring two different alloy steels to the same tensile strength. Figure
921 shows this relationship between the mechanical properties of steels
with tempered martensitic structures. The expected scatter in values is
indicated by the shading. Because of this similarity in properties, it is
logical to ask why so many different alloy steels are used. Actually, as
will be seen in Chap. 14, all lowalloy steels do not have the same impact
resistance or notch sensitivity, and they may differ considerably in these
' E. J. Janitsky and M. Baeyertz, "Metals Handbook," pp. 515518, American
Society for Metals, Metals Park, Ohio, 1939.
2 W. r.. Patton, ^Utal Prngr., vol. 43, pp. 726733, 1943.
268
Applications to Materials Testing
[Ch
ap.
respects when heattreated to tensile strengths in excess of 200,000 psi.
Further, to minimize processing difficulties such as quench cracking and
weld embrittlement, it is an advantage to use a steel with the lowest
carbon content consistent with the required asquenched hardness. For
this reason, steels are available with closely spaced carbon contents.
.400
300
200
70
60
■S 50
e 40
30
1 1 1 1 1
inS>
Brine// hardness number.
N^
^^<^
x^M<^^
^
.\S>^^^
o.\S>^^^^
v^^^
^^
^'
N^
<>
s>>^
O
N^
^,
^
^'
sS>^
N^^^
180 o
o
■ S
i>^
\''
.^^^^
^
v'^'
^:5^^
ISO °
^
.^J
^
\\>^^^
'
^
x^
^
^^Yie/d stress
i^
N^^^
120 :p
iN"^
^^^
b^
^^^
100 '>■
^^'
5N
Lowa//oy stee/s confaininc,
''<'',
'{///
'///.
and tempered ~^
>SJ
y/A
////
V/A
^//'
''///.
////
////
7/A
<///
>/A
'/,
'
'^yy
y//
y/A
/.,
^///
y/y
/?%
yy/
////
^/.,
'^//.
Y//
y/^
yy/.
'///,
'^//
////
•^6
y//
^^''
'//^
^yp/
yy?
y//.
'///
Reduction of are
n^
' '//
vy/
/v/^
'yyy}
vy/
30 c
o
'^^,
yy/^
^/.
'y^y
'V^.
'//^
«<C
y/yA
a>
^^
'///.
o
'''/?
'^^
/ ' ■
^E/onaafinn
20 <u
/^
f>^
'(.'''■
'^//
^/y^
<///
'/^
YM
y//^
y//.
a>
'■'^,
'yyy
^/7
^
'''//
in ^
12
14
Ten
sile s
tren
gth,
IS
,00(
Dpsi
2C
)0
Fig. 921 . Relationships between tensile properties of quenched and tempered low
alloy steels. iyV. G. Patton, Metal Progr., vol. 43, p. 726, 1943.)
Steel sections which are too large to be quenched throughout to essen
tially 100 per cent martensite will contain highertemperature transfor
mation products such as ferrite, pearlite, and bainite interspersed with
the martensite. Such a situation is known as a slackquenched structure.
Slack quenching results in tensile properties which are somewhat poorer
than those obtained with a completely tempered martensitic structure.
The yield strength and the reduction of area are generally most affected,
Sec. 911] The Tension Test 269
while impact strength can be very greatly reduced. The effect of slack
quenching is greatest at high hardness levels. As the tempering temper
ature is increased, the deviation in the properties of slackquenched steel
from those of tempered martensite becomes smaller. In steels with suf
ficient hardenability to form 100 per cent martensite it is frequently found
that not all the austenite is transformed to martensite on quenching.
Studies' have shown that the greatest effect of retained austenite on
tensile properties is in decreasing the yield strength.
91 1 . Anisotropy of Tensile Properties
It is frequently found that the tensile properties of wroughtmetal
products are not the same in all directions. The dependence of proper
ties on orientation is called anisotropy. Two general types of anisotropy
are found in metals. Crystallographic anisotropy results from the pre
ferred orientation of the grains which is produced by severe deformation.
Since the strength of a single crystal is highly anisotropic, a severe plastic
deformation which produces a strong preferred orientation will cause a
polycrystalline specimen to approach the anisotropy of a single crystal.
The yield strength, and to a lesser extent the tensile strength, are the
properties most affected. The yield strength in the direction perpen
dicular to the main (longitudinal) direction of working may be greater
or less than the yield strength in the longitudinal direction, depending
upon the type of preferred orientation which exists. This type of ani
sotropy is most frequently found in nonferrous metals, especially when
they have been severely worked into sheet. Crystallographic anisotropy
can be eliminated by recrystallization, although the formation of a
recrystallization texture can cause the reappearance of a different type
of anisotropy. A practical manifestation of crystallographic anisotropy
is the formation of "ears," or nonuniform deformation in deepdrawn
cups. Crystallographic anisotropy may also result in the elliptical defor
mation of a tensile specimen.
Mechanical fihering is due to the preferred alignment of structural dis
continuities such as inclusions, voids, segregation, and second phases in
the direction of working. This type of anisotropy is important in forg
ings and plates. The principal direction of working is defined as the
longitudinal direction. This is the long axis of a bar or the rolling direc
tion in a sheet or plate. Two transverse directions must be considered.
The shorttransverse direction is the minimum dimension of the product,
for example, the thickness of a plate. The longtransverse direction is
perpendicular to both the longitudinal and shorttransverse directions.
> L. S. Castleman, B. L. Averbach, and M. Cohen, Trans. ASM, vol. 44, pp. 240
263, 1952.
270
Applications to Materials Testing
[Ch
ap.
In a round or square, both these transverse directions are equivalent,
while in a sheet the properties in the shorttransverse direction cannot be
measured. In wroughtsteel products mechanical fibering is the princi
pal cause of directional properties. Measures of ductility like reduction
of area are most affected. In general, reduction of area is lowest in the
shorttransverse direction, intermediate in the longtransverse direction,
and highest in the longitudinal direction.
70
Lo
ngitud
inal
Transverse
\
1
1 1
1
1
_
:
55 60
D
o50
Max
:
^
_
.40

\.Min

T3

(^u
= 166,000 psi

.^30

(^0
= 146,000 psi
V
^^ 
20
1
1 1
1
1
20
40 60
Angle, deg
Fig. 922. Relationship between reduction of area and angle between the longitudinal;
direction in forging and the specimen axis. (C. Wells and R. F. Mehl, Trans. ASM, vol.
41, p. 753, 1949.)
Transversa properties are particularly important in thickwalled tubes,
like guns and pressure vessels, which are subjected to high internal pres
sures. In these applications the greatest principal stress acts in the cir
cumferential direction, which corresponds to the transverse direction of
a cylindrical forging. "While there is no direct method for incorporating
the reduction of area into the design of such a member, it is known that
the transverse reduction of area (RAT) is a good index of steel quality
for these types of applications. For this reason the RAT may be the
limiting value in the design of a part. A great deal of work^""^ on the
transverse properties of gun tubes and large forgings has provided data
in this field. Figure 922 shows the variation of reduction of area with
the angle between the axis of the tensile specimen and the longitudinal
direction in a forging of SAE 4340 steel. No similar variation with
orientation is found for the yield strength or the tensile strength. This
figure shows both the maximum and minimum values of reduction of
area obtained for different specimen orientations. Because of the large
1 C. Wells and R. F. Mehl, Trans. ASM, vol. 41, pp. 715818, 1949.
2 A. H. Grobe, C. Wells, and R. F. Mehl, Trans. ASM, vol. 45, pp. 10801122, 1963.
' E. A. Loria, Trans. ASM, vol. 42, pp. 486498, 1950.
Sec. 911]
TheTc
Test
271
scatter in measurements of RAT it is necessary to use statistical methods.
The degree of anisotropy in reduction of area increases with strength
level. In the region of tensile strength between 80,000 and 180,000 psi
the RAT decreases by about 1.5 per cent for each 5,000 psi increase in
tensile strength. Figure 923 shows the way in which the longitudinal
and transverse reduction of area varies with reduction by forging. The
forging ratio is the ratio of the original to the final crosssectional area
72
1
1 1 1
68
^6A
'
L ongitudinal
:/
r
O
 /
<u
/
° 60

/
o
 /
t=
/
.5 56
 /^
■\
o
3
■XD
\
,2 52

\3\

48
_
x^^
.
x^
46
" 1
1 1 1 1 r 1
l:l 3:1 5:1 7:1
Forging ratio
Fig. 923. Effect of forging reduction on longitudinal and transverse reduction of
area. Tensile strength 118,000 psi. (C. Wells and R. F. Mehl, Trans. ASM, vol. 41,
p. 755, 1949.)
of the forging. It is usually found that the optimum properties arc
obtained with a forging ratio of 2 to 3:1. Nonmetallic inclusions are
considered to be a major source of low transverse ductility. This is
based on the fact that vacuummelted steels give higher RAT values
than airmelted steel and on correlations which have been made' between
inclusion content and RAT. Other factors such as microsegregation and
dendritic structure may also be responsible for low transverse ductility in
forgings.
An interesting aspect of the anisotropic strength of metals concerns
the effect of prior torsional deformation on tensile properties. Swift^
twisted mildsteel bars in torsion and then determined the tensile proper
tias of the bar. If the torsional shear strain at the surface exceeded
unity, it was found that the tensilefracture stress and reduction of area
were greatly reduced. At the same time the tensile fracture changed
1 J. Welchner and W. G. Hildorf, Trans. ASM, vol. 42, pp. 455485, 1950.
2 H. W. Swift, J. Iron Steel Inst. {London), vol. 140, p. 181, 1939.
272 Applications to Materials Testing [Chap. 9
from a cupandcone fracture to a fracture on a 45° plane. If the speci
mens were twisted to this strain and then untwisted, there was little
effect of the torsional deformation on the fracture stress, ductility, or
type of fracture. In interpreting these results it was suggested^ that the
twisting produced a preferred orientation of initially randomly oriented
microcracks. The cracks were presumed to become oriented along the
helical surface, which is in compression during twisting (see Fig. 104).
Separation occurs along this 45° plane when axial tension is applied. The
cracks were assumed to become reoriented in the longitudinal direction
of the bar when it was untwisted, and with this orientation they would
have little effect on the tensile properties. Although there was no real
experimental evidence for the existence of microcracks, it was considered
that they could be initiated at inclusions and secondphase particles.
However, similar experiments^ with OFHC copper, in which there were
no secondphase particles and no preferred orientation, confirmed and
extended Swift's observations. The mechanical anisotropy which was
observed was explained on the assumption that the metal contained a
fibrous flaw structure, which has the characteristics of submicroscopic
cracks. There is some indication that these cracks originate during
solidification of the ingot and perhaps during plastic deformation, when
they are oriented in the principal working direction.
BIBLIOGRAPHY
Hollomon, J. H., and L. D. Jaffee: "Ferrous Metallurgical Design," chaps. 3 and 4,
John Wiley & Sons, Inc., New York, 1947.
Lessells, John M.: "Strength and Resistance of Metals," chap. 1, John Wiley & Sons,
Inc., New York, 1954.
Low, J. R., Jr.: Behavior of Metals under Direct or Nonreverse Loading, in "Prop
erties of Metals in Materials Engineering," American Society for Metals, Metals
Park, Ohio, 1949.
Marin, J.: "Engineering Materials," chaps. 1 and 11, Prentice Hall, Inc., Englewood
Cliffs, N.J., 1952
Nadai, A.: "Theory of Flow and Fracture of Solids," vol. I, chap. 8, McGrawHill
Book Company, Inc., New York, 1950.
Symposium on Significance of the Tension Test of Metals in Relation to Design,
Proc. ASTM, vol. 40, pp. 501609, 1940.
1 C. Zener and J. H. Hollomon, Trans. ASM, vol. 33, p. 163, 1944.
2 W. A. Backofen, A. J. Shaler, and B. B. Hundy, Trans. ASM, vol. 46, pp. 655680,
1954.
Chapter 10
THE TORSION TEST
101. Introduction
The torsion test has not met with the wide acceptance and the use
that have been given the tension test. However, it is useful in many
engineering applications and also in theoretical studies of plastic flow.
Torsion tests are made on materials to determine such properties as the
modulus of elasticity in shear, the torsional yield strength, and the
modulus of rupture. Torsion tests also may be carried out on fullsized
parts, such as shafts, axles, and twist drills, which are subjected to tor
sional loading in service. It is frequently used for testing brittle mate
rials, such as tool steels, and has been used in the form of a hightem
perature twist test to evaluate the forgeability of materials. The torsion
test has not been standardized to the same extent as the tension test
and is rarely required in materials specifications.
Torsiontesting equipment consists of a twisting head, with a chuck for
gripping the specimen and for applying the twisting moment to the speci
men, and a weighing head, which grips the other end of the specimen
and measures the twisting moment, or torque. The deformation of the
specimen is measured by a twistmeasuring device called a troptometer.
Determination is made of the angular displacement of a point near one
end of the test section of the specimen with respect to a point on the
same longitudinal element at the opposite end. A torsion specimen
generally has a circular cross section, since this represents the simplest
geometry for the calculation of the stress. Since in the elastic range
the shear stress varies linearly from a value of zero at the center of the
bar to a maximum value at the surface, it is frequently desirable to test
a thinwalled tubular specimen. This results in a nearly uniform shear
stress over the cross section of the specimen.
102. Mechanical Properties in Torsion
Consider a cylindrical bar which is subjected to a torsional moment at
one end (Fig. 101). The twisting moment is resisted by shear stresses
273
274 Applications to Materials Testing
[Chap. 10
set up in the cross section of the bar. The shear stress is zero at the
center of the bar and increases Hnearly with the radius. Equating the
Fig. 101. Torsion of a solid bar.
twisting moment to the internal resisting moment,
rr = a fa
Mt = TfdA =  r'dA
/r = o r jo
(101)
But jr"^ dA is the polar moment of inertia of the area with respect to the
axis of the shaft. Thus,
rJ
or
Mt = —
r
MtT
(102)
where r = shear stress, psi
Mt = torsional moment, Ibin.
r = radial distance measured from center of shaft, in.
J = polar moment of inertia, in.^
Since the shear stress is a maximum at the surface of the bar, for a solid
cylindrical specimen where / = 7ri)V32, the maximum shear stress is
^ MtD/2 ^ \QMt
'"  7rZ)V32 " ttD^
For a tubular specimen the shear stress on the outer surface is
^ IQMtDi
(103)
(104)
where Di = outside diameter of tube
D2 = inside diameter of tube
The troptometer is used to determine the angle of twist, d, usually
expressed in radians. If L is the test length of the specimen, from Fig.
Sec. 102J The Torsion Test 275
101 it will be seen that the shear strain is given by
rd
7 = tan = — (105)
During a torsion test measurements are made of the twisting moment Mr
and the angle of twist, 6. A torquetwist diagram is usually obtained,
as shown in Fig. 102.
The elastic properties in torsion may be obtained by using the torque
at the proportional limit or the torque at some offset angle of twist,
Offset
Angle of twist, deg
Fig. 102. Torquetwist diagram.
frequently 0.001 radian/in. of gage length, and calculating the shear
stress corresponding to the twisting moment from the appropriate equa
tions given above. A tubular specimen is usually required for a precision
measurement of the torsional elastic limit or yield strength. Because of
the stress gradient across the diameter of a solid bar the surface fibers are
restrained from yielding by the less highly stressed inner fibers. Thus,
the first onset of yielding is generally not readily apparent with the
instruments ordinarily used for measuring the angle of twist. The use
of a thinwalled tubular specimen minimizes this effect because the stress
gradient is practically eliminated. Care should be taken, however, that
the wall thickness is not reduced too greatly, or the specimen will fail by
buckling rather than torsion. Experience has shown that for determi
nations of the shearing yield strength and modulus of elasticity the ratio
of the length of the reduced test section to the outside diameter should be
about 10 and the diameterthickness ratio should be about 8 to 10.
Once the torsional yield strength has been exceeded the shearstress
distribution from the center to the surface of the specimen is no longer
linear and Eq. (103) or (104) does not strictly apply. However, an
ultimate torsional shearing strength, or modulus of rupture, is frequently
determined by substituting the maximum measured torque into these
276 Applications to Materials Testing [Chap. 10
equations. The results obtained by this procedure overestimate the
ultimate shear stress. A more precise method of calculating this value
will be discussed in the next section. Although the procedure just
described results in considerable error, for the purpose of comparing and
selecting materials it is generally sufficiently accurate. For the determi
nation of the modulus of rupture with tubular specimens, the ratio of
gage length to diameter should be about 0.5 and the diameterthickness
ratio about 10 to 12.
Within the elastic range the shear stress can be considered proportional
to the shear strain. The constant of proportionality, G, is the modulus
of elasticity in shear, or the modulus of rigidity.
T =Gy (106)
Substituting Eqs. (102) and (105) into Eq. (106) gives an expression
for the shear modulus in terms of the geometry of the specimen, the
torque, and the angle of twist.
MtL
G = ^ (107)
103. Torsional Stresses for Large Plastic Strains
Beyond the torsional yield strength the shear stress over a cross section
of the bar is no longer a linear function of the distance from the axis, and
Eqs. (103) and (104) do not apply. Nadai^ has presented a method for
calculating the shear stress in the plastic range if the torquetwist curve
is known. To simplify the analysis, we shall consider the angle of twist
per unit length, 6', where 6' = d/L. Referring to Eq. (105), the shear
strain will be
7 = rd' (108)
Equation (101), for the resisting torque in a cross section of the bar,
can be expressed as follows:
Mn
= 27r y"^" rr^ dr (109)
Now the shear stress is related to the shear strain by the stressstrain
curve in shear.
T = /(t)
Introducing this equation into Eq. (109) and changing the variable from
^ A. Nadai, "Theory of Flow and Fracture of Solids," 2d ed., vol. I, p. 347349,
McGrawHill Book Company, Inc., New York, 1950. A generalization of this
analysis for strainrate sensitive materials has been given by D. S. Fields and W. A.
Backofen, Proc. ASTM, vol. 57, pp. 12591272, 1957.
Sec. 103]
r to 7 by means of Eq. (108) gives
Mt = 27r
The!
orsion
Test
277
^^^' {d'Y e'
Mrid'Y = 27r / "f(y)Ydy (1010)
where 7a = ad'. Differentiating Eq. (1010) with respect to 6'
^ (Mre") = 2wafiad')a'id'r = 2TaHe'mae')
But, the maximum shear stress in the bar at the outer fiber is Ta = /(ad')
Therefore,
diMrd")
dd'
= 27ra»(0')V„
Therefore,
dMr
SMrier + (G'y^ = 2ira\d'YTa
(1011)
If a torquetwist curve is available, the shear stress can be calculated
with the above equation. Figure 103 illustrates how this is done.
Angle of twist per unit lengtti d
Fig. 103. Method of calculating shear stress from torquetwist diagram.
Examination of Eq. (1011) shows that it can be written in terms of the
geometry of Fig. 103 as follows:
27ra'
{BC + 3CD)
(1012)
It will also be noticed from Fig. 103 that at the maximum value of
torque dMr/dd' = 0. Therefore, the ultimate torsional shear strength
278 Applications to Materials Testing [Chap. 10
or modulus of rupture can be expressed by
3M,
r,, =
^^ (1013)
104. Types of Torsion Failures
Figure 104 illustrates the state of stress at a point on the surface of a
bar subjected to torsion. The maximum shear stress occurs on two
mutually perpendicular planes, perpendicular to the longitudinal axis
^max
'73= C^1
Fig. 104. State of stress in torsion.
(a) {i>)
Fig. 105. Typical torsion failures, (a) Shear (ductile) failure; (b) tensile (brittle)
failure.
yy and parallel with the longitudinal axis xx. The principal stresses
cTi and 03 make an angle of 45° with the longitudinal axis and are equal
in magnitude to the shear stresses, oi is a tensile stress, and 03 is
an equal compressive stress. The intermediate stress a 2 is zero.
Torsion failures are different from tensile failures in that there is little
localized reduction of area or elongation. A ductile metal fails by shear
along one of the planes of maximum shear stress. Generally the plane
of the fracture is normal to the longitudinal axis (see Fig. 105a). A
brittle material fails in torsion along a plane perpendicular to the direc
tion of the maximum tensile stress. Since this plane bisects the angle
between the two planes of maximum shear stress and makes an angle of
45° with the longitidunal and transverse directions, it results in a helical
fracture (Fig. 1056). Fractures are sometimes observed in which the
test section of the specimen breaks into a large number of fairly small
Sec. 105] The Torsion Test 279
pieces. In these cases it can usually be determined that the fracture
started on a plane of maximum shear stress parallel with the axis of the
specimen. A study of torsion failures in a tool steel as a function of
hardness^ showed that fracture started on planes of maximum shear stress
up to a Vickers hardness of 720 and that above this hardness tensile
stresses were responsible for starting fracture.
105. Torsion Test vs. Tension Test
A good case can be made for the position advanced by Sauveur^ that
the torsion test provides a more fundamental measure of the plasticity
of a metal than the tension test. For one thing, the torsion test yields
directly a shearstressshearstrain curve. This type of curve has more
fundamental significance in characterizing plastic behavior than a stress
strain curve determined in tension. Large values of strain can be
obtained in torsion without complications such as necking in tension or
barreling due to frictional end effects in compression. Moreover, in
torsion, tests can be made fairly easily at constant or high strain rates.
On the other hand, considerable labor is involved in converting torque
angleoftwist data into shearstressstrain curves. Furthermore, unless
a tubular specimen is used, there will be a steep stress gradient across
the specimen. This will make it difficult to make accurate measure
ments of the yield strength.
The tension test and the torsion test are compared below in terms of
the state of stress and strain developed in each test.
Tension test Torsion test
0"! = (Tmax', 0^2 = Cs = O"! = ~0'3', Cr2 =
0"! 0max So"!
' max
2 2 max 2 ^1
_ __«i __._n
Cmax — Cl, €2 — €3 — ~ K ^max — Cl — ^3 , ^2 — W
Q
Traax = Sinh^ Tmax = €1 — 63 = 2ei
i = [H(e,' + 62^ + e^r^^
ff = (71 & = V 3(ri
2 7
1 R. D. Olleman, E. T. Wessel, and F. C. Hull, Trans. ASM, vol. 46, pp. 8799, 1954.
2 A. Sauveur, Proc. ASTM, vol. 38, pt. 2, pp. 320, 1938.
280 Applications to Materials Testing
[Chap. 10
This comparison shows that rmax will be twice as great in torsion as in
tension for a given value of omax. Since as a first approximation it can be
considered that plastic deformation occurs on reaching a critical value of
Tmax and brittle fracture occurs on reaching a critical value of omax, the
opportunity for ductile behavior is greater in torsion than in tension.
This is illustrated schematically in Fig. 106, which can be considered
representative of the condition for a brittle material such as hardened
Critical r^ax for plastic flow
Critical
! CTmax for
fracture
Fig. 106. Effect of ratio Tmax/cmax in determining ductility. (After Gensamer.)
_
100

80

^,^"'■'0
""o
60
 /
'/
rjsjJ^S^ —
40
^
20
^
a vs. € for torsion
1 1
1 1 1 1
1 1 1 1 1 1 1
0.2 0.4 0.6 0.8 1.0 1,2 1.4
True strain
Fig. 107. Tension and torsion truestresstruestrain curves for lowcarbon steel.
Sec. 105] The Torsion Test 281
tool steel. In the torsion test the critical shear stress for plastic flow is
reached before the critical normal stress for fracture, while in tension the
critical normal stress is reached before the shear stress reaches the shear
stress for plastic flow. Even for a metal which is ductile in the tension
test, where the critical normal stress is pushed far to the right in Fig.
106, the figure shows that the amount of plastic deformation is greater
in torsion than in tension.
The tensile stressstrain curve can be derived from the curve for tor
sion when the stressstrain curve is plotted in terms of significant stress
and strain or the octahedral shear stress and strain (see Prob. 10.4).
Figure 107 shows a truestresstruestrain curve from a tension test and
the shearstressshearstrain curve for the same material in torsion.
When both curves are plotted in terms of significant stress and signifi
cant strain (the tension curve is unchanged), the two curves superimpose
within fairly close limits. A number of examples of this can be found in
the literature.*'^ Also, a straight line is obtained for torsion data when
the logarithm of significant stress is plotted against the logarithm of sig
nificant strain.^ The values of K and n obtained from these curves agree
fairly well with comparable values obtained from the tension test.
BIBLIOGRAPHY
Davis, H. E., G. E. Troxell, and C. T. Wiskocil: "The Testing and Inspection of Engi
neering Materials," chap. 5, 2d ed., McGrawHill Book Company, Inc., New
York, 1955.
Gensamer, M.: "Strength of Metals under Combined Stresses," American Society for
Metals, Metals Park, Ohio, 1941.
Marin, J.: "Engineering Materials," chap. 2, PrenticeHall, Inc., Englewood Cliffs,
N.J., 1952.
"Metals Handbook," pp. 111112, American Society for Metals, Metals Park, Ohio,
1948.
1 E. A. Davis, Trans. ASME, vol. 62, pp. 577586, 1940.
2 J. H. Faupel and J. Marin, Trans. ASM, vol. 43, pp. 9931012, 1951.
3 H. Larson and E. P. Klier, Trans. ASM, vol. 43, pp. 10331051, 1951.
Chapter 11
THE HARDNESS TEST
111. Introduction
The hardness of a material is a poorly defined term which has many
meanings depending upon the experience of the person involved. In
general, hardness usually implies a resistance to deformation, and for
metals the property is a measure of their resistance to permanent or
plastic deformation. To a person concerned with the mechanics of mate
rials testing, hardness is most likely to mean the resistance to indentation,
and to the design engineer it often means an easily measured and speci
fied quantity which indicates something about the strength and heat
treatment of the metal. There are three general types of hardness
measurements depending upon the manner in which the test is con
ducted. These are (1) scratch hardness, (2) indentation hardness, and
(3) rebound, or dynamic, hardness. Only indentation hardness is of
major engineering interest for metals.
Scratch hardness is of primary interest to mineralogists. With this
measure of hardness, various minerals and other materials are rated on
their ability to scratch one another. Hardness is measured according to
the Mohs scale. This consists of 10 standard minerals arranged in the
order of their ability to be scratched. The softest mineral in this scale
is talc (scratch hardness 1), while diamond has a hardness of 10. A
fingernail has a value of about 2, annealed copper has a value of 3, and
martensite a hardness of 7. The Mohs scale is not well suited for metals
since the intervals are not widely spaced in the highhardness range.
Most hard metals fall in the Mohs hardness range of 4 to 8. A different
type of scratchhardness test^ measures the depth or width of a scratch
made by drawing a diamond stylus across the surface under a definite
load. This is a useful tool for measuring the relative hardness of micro
constituents, but it does not lend itself to high reproducibility or extreme
accuracy.
In dynamichardness measurements the indenter is usually dropped
1 E. B. Bergsman, ASTM Bull. 176, pp. 3743, September, 1951.
282
Sec. 112] The Hardness Test 283
onto the metal surface, and the hardness is expressed as the energy of
impact. The Shore sceleroscope, which is the commonest example of a
dynamichardness tester, measures the hardness in terms of the height
of rebound of the indenter.
112. Brinell Hardness
The first widely accepted and standardized indentationhardness test
was proposed by J. A. Brinell in 1900. The Brinell hardness test con
sists in indenting the metal surface with a 10mmdiameter steel ball at
a load of 3,000 kg. For soft metals the load is reduced to 500 kg to
avoid too deep an impression, and for very hard metals a tungsten car
bide ball is used to minimize distortion of the indenter. The load is
applied for a standard time, usually 30 sec, and the diameter of the inden
tation is measured with a lowpower microscope after removal of the load.
The average of two readings of the diameter of the impression at right
angles should be made. The surface on which the indentation is made
should be relatively smooth and free from dirt or scale. The Brinell
hardness number (BHN) is expressed as the load P divided by the surface
area of the indentation. This is expressed by the formula^
BHN = ^ , (111)
(tD/2){D  VD^  d^)
where P = applied load, kg
D = diameter of ball, mm
d = diameter of indentation, mm
It will be noticed that the units of the BHN are kilograms per square
millimeter. However, the BHN is not a satisfactory physical concept
since Eq. (111) does not give the mean pressure over the surface of the
indentation.
In general, the Brinell hardness number of a material is constant only
for one applied load and diameter of ball. It has been shown that in
order to obtain the same Brinell hardness number at a nonstandard load
geometrical similitude must be maintained. This requires that the ratio
of the indentation to the indenter, d/ D, remains constant. To a first
approximation this can be attained when P ID^ is kept constant.
The greatest error in Brinell hardness measurements occurs in measur
ing the diameter of the impression. It is assumed that the diameter of
the indentation is the same as the diameter when the ball was in con
tact with the metal. However, owing to elastic recovery, the radius of
curvature of the indentation will be larger than that of the spherical
^ Tables giving BHN as a function of d for standard loads may be found in most of
the references in the Bibliography at the end of this chapter.
284 Applications to Materials Testing
[Chap. 11
indenter, although the indentation will still be symmetrical. The harder
the metal, the greater the elastic recovery. Elastic recovery will affect
measurements of the depth of indentation, but it will have only a negli
gible effect on the chordal diameter of the impression, so that this does
not in general influence Brinell hardness. However, two types of anoma
lous behavior can occur as a result of localized deformation of the metal
at the indentation. These are shown schemat
ically in cross section through the indentation
in Fig. 111. The sketch at the top illustrates
"ridging," or ''piling up," in which a lip of metal
forms around the edge of the impression. This
behavior is most common in coldworked metals
with little ability to strainharden. The measured
diameter is greater than the true diameter of the
impression, but since the ridge carries part of the
load, it is customary to base the hardness measure
ment on the diameter d shown in the sketch. The
drawing on the bottom shows ''sinking in," in
which there is a depression of the metal at the rim
of the indentation. This type of behavior is com
mon with annealed metals having a high rate of
strain hardening. The true diameter of the impres
sion can sometimes be obtained by coating the ball with bluing or dye
before making the indentation. It is frequently desirable to increase
the sharpness of definition of the impression so that the diameter can be
measured more accurately. This can sometimes be done by using a
lightly etched steel ball or by coating the surface with a dull black
pigment.
Fig. 111. Cross sections
through Brinell inden
tations illustrating (o)
ridging and (6) sinking
in.
113. Meyer Hardness
Meyer^ suggested that a more rational definition of hardness than that
proposed by Brinell would be one based on the 'projected area of the impres
sion rather than the surface area. The mean pressure between the sur
face of the indenter and the indentation is equal to the load divided by
the projected area of the indentation.
Meyer proposed that this mean pressure should be taken as the measure
of hardness. It is referred to as the Meyer hardness.
4P
Meyer hardness = —r^
(112)
1 E. Meyer, Z. Ver. deut. Ing, vol. 52, pp. 645654, 1908.
Sec. 113] The Hardness Test 285
Like the Brinell hardness, Meyer hardness has units of kilograms per
square millimeter. The Meyer hardness is less sensitive to the applied
load than the Brinell hardness. For a coldworked material the Meyer
hardness is essentially constant and independent of load, while the Brinell
hardness decreases as the load increases. For an annealed metal the
Meyer hardness increases continuously with the load because of strain
hardening produced by the indentation. The Brinell hardness, however,
first increases with load and then decreases for still higher loads. The
Meyer hardness is a more fundamental measure of indentation hardness ;
yet it is rarely used for practical hardness measurements.
Meyer proposed an empirical relation between the load and the size of
the indentation. This relationship is usually called Meyer's law.
P = kd'^' (113)
where P = applied load, kg
d = diameter of indentation, mm
n' = a material constant related to strain hardening of metal
fc = a material constant expressing resistance of metal to
penetration
The parameter n' is the slope of the straight line obtained when log P is
plotted against log d, and k is the value of P at c? = 1. Fully annealed
metals have a value of n' of about 2.5, while n' is approximately 2 for
fully strainhardened metals. This parameter is roughly related to the
strainhardening coefficient in the exponential equation for the true
stresstruestrain curve. The exponent in Meyer's law is approximately
equal to the strainhardening coefficient plus 2.
When indentations are made with balls of different diameters, different
values of k and n' will be obtained.
P = A:ii)i"'' = k,D2"^' = ksDs"^' ■ ' '
Meyer found that n' was almost independent of the diameter of the
indenter D but that k decreased with increasing values of D. This can
be expressed empirically by a relationship of the form
C = kiDi''~ = A:2/)2"' = ksDs'^ ■ ■ •
The general expression for Meyer's law then becomes
^ Cdi"' ^ Cd^ ^ Cd^'^'
Z)i"'~2 i)2"'2 Z)3"'2 ^^^~^'
Several interesting conclusions result from Eq. (114). First, this equa
tion can be written
(115)
286 Applications to Materials Testing [Chap. 1 1
Since dl D must be constant for geometrically similar indentations, the
ratio F Idr must also be constant. However, F jd} is proportional to the
Meyer hardness. Therefore, geometrically similar indentations give the
same Meyer hardness number. Equation (114) can also be rearranged
to give
(116)
Remembering again that geometrically similar indentations are obtained
when d/D is constant, we see that the above equation shows that the
ratio P/D^ must also provide the same result. Therefore, the same
hardness values will be obtained when the ratio F/D^ is kept constant.
There is a lower limit of load below which Meyer's law is not valid.
If the load is too small, the deformation around the indentation is not
fully plastic and Eq. (113) is not obeyed. This load will depend upon
the hardness of the metal. For a 10mmdiameter ball the load should
exceed 50 kg for copper with a BHN of 100, and for steel with a BHN
of 400 the load should exceed 1,500 kg. For balls of different diameter
the critical loads will be proportional to the square of the diameter.
1'4. Analysis of Indentation by a Spherical Indenter
Tabor^ has given a detailed discussion of the mechanics of deformation
of a flat metal surface with a spherical indenter. The elements of this
analysis will be described here. Figure 112 illustrates the process. For
•0
(a)
Fig. 112. Plastic deformation of an ideal plastic material by a spherical indenter.
(a) Beginning of plastic deformation at point 0; (b) full plastic flow. {After D. Tabor,
"The Hardness of Metals," p. 47, Oxford University Press, New York, 1951.)
an ideal plastic metal with no strain hardening the highest pressure occurs
immediately below the surface of contact at a depth of about d/2. The
pressure at this point is about 0A7pm, where pm is the mean pressure over
the circle of contact. Assuming that the maximumshearstress theory is
» D. Tabor, "The Hardness of Metals," Oxford University Press, New York, 1951.
Sec. 115] The Hardness Test 287
the criterion for plastic flow, we can write
0A7pm = 0.5(To
or Pm « l.loo (117)
where oo is the yield stress in tension or compression.
Therefore, the deformation under the indenter is elastic until the mean
pressure reaches about 1.1 times the yield stress. At about this pressure
plastic deformation begins in the vicinity of point (Fig. ll2a). As the
load is further increased, the mean pressure increases and the plastically
deformed region grows until it contains the entire region of contact (Fig.
1126). An analytical solution for the pressure between the spherical
indenter and the indentation under conditions of full plasticity is very
difficult. The best analysis of this problem indicates that pm « 2.66cro.
Meyer hardness tests on severely coldworked metal indicates that full
plasticity occurs when
Pm « 2.8(ro (118)
For an ideally plastic metal the pressure would remain constant at this
value if the load were increased further. Since real metals strainharden,
the pressure would increase owing to an increase in oo as the indentation
process was continued. Most Brinell hardness tests are carried out
under conditions where full plasticity is reached. This is also a neces
sary condition for Meyer's law to be valid.
115. Relationship between Hardness and the TensileHow Curve
Tabor^ has suggested a method by which the plastic region of the
true stressstrain curve may be determined from indentation hardness
measurements. The method is based on the fact that there is a simi
larity in the shape of the flow curve and the curve obtained when the
Meyer hardness is measured on a number of specimens subjected to
increasing amounts of plastic strain. The method is basically empirical,
since the complex stress distribution at the hardness indentation pre
cludes a straightforward relationship with the stress distribution in the
tension test. However, the method has been shown to give good agree
ment for several metals and thus should be of interest as a means of
obtaining flow data in situations where it is not possible to measure
tensile properties. The true stress (flow stress) is obtained from Eq.
(118), where co is to be considered the flow stress at a given value of
true strain.
Meyer hardness = Pm = 2.8cro
From a study of the deformation at indentations. Tabor concluded that
1 Tabor, op. cit., pp. 6776; /. Inst. Metals, vol. 79, p. 1, 1951.
288 Applications to Materials Testing [Chap. 11
the true strain was proportional to the ratio d/ D and could be expressed as
d
e = 0.2
D
(119)
Thus, if the Meyer hardness is measured under conditions such that d/D
varies from the smallest value for full plasticity up to large values and
Eqs. (118) and (119) are used, it
•"" ' is possible at least to approximate
the tensilefiow curve. Figure 113
shows the agreement which has
been obtained by Tabor between
the flow curve and hardness versus
d/D curve for mild steel and an
nealed copper. Tabor's results
have been verified by Lenhart^
for duralumin and OFHC copper.
However, Tabor's analysis did not
predict the flow curve for mag
nesium, which was attributed by
Lenhart to the high anisotropy of
deformation in this metal. This
work should not detract from the
usefulness of this correlation but,
rather, should serve to emphasize
that its limitations should be in
vestigated for new applications.
There is a very useful engineer
ing correlation between the Brinell hardness and the ultimate tensile
strength of heattreated plaincarbon and mediumalloy steels.
Ultimate tensile strength, in pounds per square inch, = 500(BHN).
A brief consideration will show that this is in agreement with Tabor's
results. If we make the simplifying assumption that this class of mate
rials does not strainharden, then the tensile strength is equal to the
yield stress and Eq. (118) applies.
(Tu = TynVm = 0.36p„j kg/mm^
The Brinell hardness will be only a few per cent less than the value of
Meyer hardness p^. Upon converting to engineering units the expression
becomes
au = 515(BHN)
Fig. 1 1 3. Comparison of flow curve deter
mined from hardness measurements
(circles, crosses) with flow curve deter
mined from compression test (solid lines).
{D. Tabor, "The Haidness of Metals," p.
74, Oxford University Press, New York,
1951.)
1 R. E. Lenhart, WADC Tech. Rept. 55114, June, 1955.
>cc.
116] The Hardness Test 289
It shovild now be apparent why the same relationship does not hold for
other metals. P'or example, for annealed copper the assumption that
strain hardening can be neglected will be grossly in error. For a metal
with greater capability for strain hardening the "constant" of propor
tionality will be greater than that used for heattreated steel.
116. Vickers Hardness
The Vickers hardness test uses a squarebase diamond pyramid as the
indenter. The included angle between opposite faces of the pyramid is
136°. This angle was chosen because it approximates the most desirable
ratio of indentation diameter to ball diameter in the Brinell hardness test.
Because of the shape of the indenter this is frequently called the diamond
pyramid hardness test. The diamondpyramid hardness number (DPH),
or Vickers hardness number (VHN, or VPH), is defined as the load
divided by the surface area of the indentation. In practice, this area is
calculated from microscopic measurements of the lengths of the diagonals
of the impression. The DPH may be determined from the following
equation,
DPH = ^^ %W2) ^ 1^ („.,0)
where P = applied load, kg
L = average length of diagonals, mm
6 = angle between opposite faces of diamond = 136"
The Vickers hardness test has received fairly wide acceptance for
research work because it provides a continuous scale of hardness, for a
given load, from very soft metals with a DPH of 5 to extremely hard
materials with a DPH of 1,500. With the Rockwell hardness test,
described in the next section, or the Brinell hardness test, it is usually
necessary to change either the load or the indenter at some point in the
hardness scale, so that measurements at one extreme of the scale cannot
be strictly compared with those at the other end. Because the impres
sions made by the pyramid indenter are geometrically similar no matter
what their size, the DPH should be independent of load. This is gener
ally found to be the case, except at very light loads. The loads ordinarily
used with this test range from 1 to 120 kg, depending on the hardness of
the metal to be tested. In spite of these advantages, the Vickers hard
ness test has not been widely accepted for routine testing because it is
slow, requires careful surface preparation of the specimen, and allows
greater chance for personal error in the determination of the diagonal
length.
A perfect indentation made with a perfect diamondpyramid indenter
[a)
290 Applications to Materials Testing [Chap. 1 1
would be a square. However, anomalies corresponding to those described
earlier for Brinell impressions are frequently observed with a pyramid
indenter (Fig. 114). The pincushion indentation in Fig. 1146 is the
result of sinking in of the metal
around the fiat faces of the pyra
mid. This condition is observed
with annealed metals and results
in an overestimate of the diagonal
length. The barrelshaped inden
Fig. 1 14. Types of diamondpyramid in ^^^:^^^ ^^ Ylg. 1 l4c is found in cold
dentations, (a) Perfect indentation ; Vb) , , , , t, ^, r
,• i,, i . i worked metals. It results from
pincushion indentation due to sinking in ;
(c) barreled indentation due to ridging. ridging or piling up of the metal
around the faces of the indenter.
The diagonal measurement in this case produces a low value of the con
tr.ct area so that the hardness numbers are erroneously high. Empirical
corrections for this effect have been proposed.^
117. Rockwell Hardness Test
The most widely used hardness test in the United States is the Rock
well hardness test. Its general acceptance is due to its speed, freedom
from personal error, ability to distinguish small hardness differences in
hardened steel, and the small size of the indentation, so that finished
heattreated parts can be tested without damage. This test utilizes the
depth of indentation, under constant load, as a measure of hardness. A
minor load of 10 kg is first applied to seat the specimen. This minimizes
the amount of surface preparation needed and reduces the tendency for
ridging or sinking in by the indenter. The major load is then applied,
and the depth of indentation is automatically recorded on a dial gage in
terms of arbitrary hardness numbers. The dial contains 100 divisions,
each division representing a penetration of 0.00008 in. The dial is
reversed so that a high hardness, which corresponds to a small penetra
tion, results in a high hardness number. This is in agreement with the
other hardness numbers described previously, but unlike the Brinell and
Vickers hardness designations, which have units of kilograms per square
miUimeter, the Rockwell hardness numbers are purely arbitrary.
One combination of load and indenter will not produce satisfactory
results for materials with a wide range of hardness A 120° diamond
cone with a slightly rounded point, called a Brale indenter, and }/{& and
3/^in. diameter steel balls are generally used as indenters. Major loads
of 60, 100, and 150 kg are used. Since the Rockwell hardness is depend
ent on the load and indenter, it is necessary to specify the combination
1 T. B. Crowe and J. F. Hinsley, /. Inst. Metals, vol. 72, p. 14, 1946.
>ec.
118] The Hardness Test 291
which is used. This is done by prefixing the hardness number with a
letter indicating the particular combination of load and indenter for the
hardness scale employed. A Rockwell hardness number without the
letter prefix is meaningless. Hardened steel is tested on the C scale with
the diamond indenter and a 150kg major load. The useful range for
this scale is from about Re 20 to Re 70. Softer materials are usually
tested on the B scale with a 3^i6in. diameter steel ball and a 100kg
major load. The range of this scale is from Rb to Rb 100. The A scale
(diamond penetrator, 60kg major load) provides the most extended
Rockwell hardness scale, which is usable for materials from annealed
brass to cemented carbides. Many other scales are available for special
purposes.^
The Rockwell hardness test is a very useful and reproducible one pro
vided that a number of simple precautions are observed. Most of the
points listed below apply ecjually well to the other hardness tests :
1. The indenter and anvil should be clean and well seated.
2. The surface to be tested should be clean, dry, smooth, and free from
oxide. A roughground surface is usually adequate for the Rockwell test.
3. The surface should be flat and perpendicular to the indenter.
4. Tests on cylindrical surfaces will give low readings, the error depend
ing on the curvature, load, indenter, and hardness of the material. Theo
retieal^ and empirical^ corrections for this effect have been published.
5. The thickness of the specimen should be such that a mark or bulge
is not produced on the reverse side of the piece. It is recommended that
the thickness be at least ten times the depth of the indentation. Tests
should be made on only a single thickness of material.
6. The spacing between indentations should be three to five times the
diameter of the indentation.
7. The speed of application of the load should be standardized. This
is done by adjusting the dashpot on the Rockwell tester. Variations in
hardness can be appreciable in very soft materials unless the rate of load
application is carefully controlled. For such materials the operating
handle of the Rockwell tester should be brought back as soon as the
major load has been fully applied.
1 1 8. Microhardncss Tests
Many metallurgical problems require the determination of hardness
over very small areas. The measurement of the hardness gradient at a
carburized surface, the determination of the hardness of individual con
» See ASTM Standard El 8.
2 W. E. Ingerson, Proc. ASTM, vol. 39, pp. 12811291, 1939.
3 R. S. Sutton and R. H. Heyer, ASTM Bull. 193, pp. 4041, October, 1953.
292 Applications to Materials Testing [Chap. 11
stituents of a microstructure, or the checking of the hardness of a delicate
watch gear might be typical problems. The use of a scratchhardness
test for these purposes was mentioned earlier, but an indentationhardness
test has been found to be more useful.^ The development of the Knoop
indenter by the National Bureau of Standards and the introduction of
the Tukon tester for the controlled application of loads down to 25 g
have made microhardness testing a routine laboratory procedure.
The Knoop indenter is a diamond ground to a pyramidal form that
produces a diamondshaped indentation with the long and short diagonals
in the approximate ratio of 7: 1. The depth of indentation is about one
thirtieth of the length of the longer diagonal. The Knoop hardness num
ber (KHN) is the applied load divided by the unrecovered projected area
of the indentation.
™N = £ = ^ (llU)
where P = applied load, kg
Ap — unrecovered projected area of indentation, mm^
L = length of long diagonal, mm
C = a constant for each indenter supplied by manufacturer
The low load used with microhardness tests requires that extreme care
be taken in all stages of testing. The surface of the specimen must be
carefully prepared. Metallographic polishing is usually required. Work
hardening of the surface during polishing can influence the results. The
long diagonal of the Knoop impression is essentially unaffected by elastic
recovery for loads greater than about 300 g. However, for lighter loads
the small amount of elastic recovery becomes appreciable. Further,
with the very small indentations produced at light loads the error in
locating the actual ends of the indentation become greater. Both these
factors have the effect of giving a high hardness reading, so that it is
usually observed that the Knoop hardness number increases as the load
is decreased below about 300 g. Tarasov and Thibault have shown that
if corrections are made for elastic recovery and visual acuity the Knoop
hardness number is constant with load down to 100 g.
119. Hardnessconversion Relationships
From a practical standpoint it is important to be able to convert the
results of one type of hardness test into those of a different test. Since
^ For a review of microhardness testing see H. Buckle, Met. Reviews, vol. 4, no. 3,
pp. 49100, 1959.
2 L. P. Tarasov and N. W. Thibault, Trans. ASM, vol. 38, pp. 331353, 1947.
Sec. 1110] The Hardness Test 293
a hardness test does not measure a welldefined property of a material
and since all the tests in common use are not based on the same type of
measurements, it is not surprising that no universal hardnessconversion
relationships have been developed. It is important to realize that hard
ness conversions are empirical relationships. The most reliable hardness
conversion data exist for steel which is harder than 240 Brinell. The
ASTM, ASM, and SAE (Society of Automotive Engineers) have agreed
on a table ^ for conversion between Rockwell, Brinell, and diamond
pyramid hardness which is applicable to heattreated carbon and alloy
steel and to almost all alloy constructional steels and tool steels in the
asforged, annealed, normalized, and quenched and tempered conditions.
However, different conversion tables are required for materials with
greatly different elastic moduli, such as tungsten carbide, or with greater
strainhardening capacity. Heyer^ has shown that the indentation hard
ness of soft metals depends on the strainhardening behavior of the mate
rial during the test, which in turn is dependent on the previous degree of
strain hardening of the material before the test. As an extreme example
of the care which is required in using conversion charts for soft metals,
it is possible for Armco iron and coldrolled aluminum each to have a
Brinell hardness of 66; yet the former has a Rockwell B hardness of 31
compared with a hardness of Rb 7 for the coldworked aluminum. On
the other hand, metals such as yellow brass and lowcarbon sheet steel
have a wellbehaved BrinellRockwell conversion^* relationship for all
degrees of strain hardening. Special hardnessconversion tables for cold
worked aluminum, copper, and 188 stainless steel are given in the ASM
Metals Handbook.
1110. Hardness at Elevated Temperatures
Interest in measuring the hardness of metals at elevated temperatures
has been accelerated by the great effort which has gone into developing
alloys with improved hightemperature strength. Hot hardness gives a
good indication of the potential usefulness of an alloy for hightemper
ature strength applications. Some degree of success has been obtained
in correlating hot hardness with hightemperature strength properties.
This will be discussed in Chap. 13. Hothardness testers using a Vickers
indenter made of sapphire and with provisions for testing in either
1 This table may be found in ASTM Standard E4847, SAE Handbook, ASM
Metals Handbook, and many other standard references.
2 R. H. Heyer, Proc. ASTM, vol. 44, p. 1027, 1944.
3 The Wilson Mechanical Instrument Co. Chart 38 for metals softer than BHN 240
(see ASM Handbook, 1948 ed., p. 101) is based on tests on these metals.
294 Applications to Materials Testing
[Chap. 11
vacuum or an inert atmosphere have been developed/ and a hightem
perature microhardness test has been described.
In an extensive review of hardness data at different temperatures
400 8 00
Temperature, °K
Fig. 115. Temperature dependence of the hardness of copper. (/. H. Westbrook,
Trans. ASM, vol. 45, p. 233, 1953.)
Westbrook^ showed that the temperature dependence of hardness could
be expressed by
H = Aexp (BT) (1112)
where H = hardness, kg/mm^
T = test temperature, °K
A, B = constants
Plots of log H versus temperature for pure metals generally yield two
straight lines of different slope. The change in slope occurs at a tem
1 F. Garofalo, P. R. Malenock, and G. V. Smith, Trans. ASM, vol. 45, pp. 377396,
1953; M. Semchyshen and C. S. Torgerson, Trans. ASM, vol. 50, pp. 830837, 1958.
2 J. H. Westbrook, Proc. ASTM, vol. 57, pp. 873897, 1957; ASTM Bull. 246,
pp. 5358, 1960.
3 J. H. Westbrook, Trans. ASM, vol. 45, pp. 221248, 1953.
Sec. 1110] The Hardness Test 295
perature which is about onehalf the melting point of the metal being
tested. Similar behavior is found in plots of the logarithm of the tensile
strength against temperature. Figure 115 shows this behavior for
copper. It is likely that this change in slope is due to a change in the
deformation mechanism at higher temperature. The constant A derived
from the lowtemperature branch of the curve can be considered to be
the intrinsic hardness of the metal, that is, H at 0°K. This value would
be expected to be a measure of the inherent strength of the binding forces
of the lattice. Westbrook correlated values of A for different metals
with the heat content of the liquid metal at the melting point and with
the melting point. This correlation was sensitive to crystal structure.
The constant B, derived from the slope of the curve, is the temperature
coefficient of hardness. This constant was related in a rather complex
way to the rate of change of heat content with increasing temperature.
With these correlations it is possible to calculate fairly well the hardness
of a pure metal as a function of temperature up to about onehalf its
melting point.
Hardness measurements as a function of temperature will show an
abrupt change at the temperature at which an allotropic transformation
occurs. Hothardness tests on Co, Fe, Ti, U, and Zr have shown^ that
the bodycentered cubic lattice is always the softer structure when it is
involved in an allotropic transformation. The facecentered cubic and
hexagonal closepacked lattices have approximately the same strength,
while highly complex crystal structures give even higher hardness. These
results are in agreement with the fact that austenitic ironbase alloys have
better hightemperature strength than ferritic alloys.
BIBLIOGRAPHY
Hardness Tests, "Metals Handbook," pp. 93105, American Society for Metals,
Metals Park, Ohio, 1948.
Lysaght, V. E.: "Indentation Hardness Testing," Reinhold Publishing Corporation,
New York, 1949.
Mott, B. W.: "Microindentation Hardness Testing," Butterworth & Co. (Publishers)
Ltd., London, 1956.
Tabor, D.: "The Hardness of Metals," Oxford University Press, New York, 1951.
Symposium on the Significance of the Hardness Test of Metals in Relation to Design,
Proc. ASTM, vol. 43, pp. 803856, 1943.
1 W. Chubb, Trans. AIME, vol. 203, pp. 189192, 1955.
Chapter 12
FATIGUE OF METALS
121. Introduction
It has been recognized since 1850 that a metal subjected to a repetitive
■or fluctuating stress will fail at a stress much lower than that required to
cause fracture on a single application of load. Failures occurring under
conditions of dynamic loading are called fatigue failures, presumably
because it is generally observed that these failures occur only after a
considerable period of service. For a long time the notion persisted that
fatigue was due to "crystallization" of the metal, but this view can no
longer be considered in the light of concepts which hold that a metal is
crystalline from the time of solidification from the melt. In fact, there
is no obvious change in the structure of a metal which has failed in
fatigue which can serve as a clue to our understanding of the reasons for
fatigue failure. Fatigue has become progressively more prevalent as
technology has developed a greater amount of equipment, such as auto
mobiles, aircraft, compressors, pumps, turbines, etc., subject to repeated
loading and vibration, until today it is often stated that fatigue accounts
for at least 90 per cent of all service failures due to mechanical causes.
A fatigue failure is particularly insidious, because it occurs without
any obvious warning. Fatigue results in a brittle fracture, with no gross
deformation at the fracture. On a macroscopic scale the fracture surface
is usually normal to the direction of the principal tensile stress. A fatigue
failure can usually be recognized from the appearance of the fracture sur
face, which shows a smooth region, due to the rubbing action as the crack
propagated through the section (top portion of Fig. 121), and a rough
region, where the member has failed in a ductile manner when the cross
section was no longer able to carry the load. Frequently the progress
of the fracture is indicated by a series of rings, or "beach marks," pro
gressing inward from the point of initiation of the failure. Figure 121
also illustrates another characteristic of fatigue, namely, that a failure
usually occurs at a point of stress concentration such as a sharp corner or
notch or at a metallurgical stress concentration like an inclusion.
Three basic factors are necessary to cause fatigue failure. These are
296
Sec. 122]
Fatigue of Metals 297
(1) a maximum tensile stress of sufficiently high value, (2) a large enough
variation or fluctuation in the applied stress, and (3) a sufficiently large
number of cycles of the applied stress. In addition, there are a host of
other variables, such as stress concentration, corrosion, temperature,
overload, metallurgical structure, residual stresses, and combined stresses,
■ o
rrri.
1
a
^
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^
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E
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Fig. 121. Fracture surface of fatigue failure which started at sharp corner of a key way
in a shaft. 1 X.
which tend to alter the conditions for fatigue. Since we have not yet
gained a basic understanding of what causes fatigue in metals, it will be
necessary to discuss each of these factors from an essentially empirical
standpoint. Because of the mass of data of this type, it will be possible
to describe only the highlights of the relationship between these factors
and fatigue. For more complete details the reader is referred to the
number of excellent publications listed at the end of this chapter.
1 22. Stress Cycles
At the outset it will be advantageous to define briefly the general types
of fluctuating stresses which can cause fatigue. Figure 122 serves to
298 Applications to Materials Testing
[Chap. 12
illustrate typical fatigue stress cycles. Figure 122a illustrates a com
pletely reversed cycle of stress of sinusoidal form. This is an idealized
situation which is produced by an R. R. Moore rotatingbeam fatigue
machine^ and which is approached in service by a rotating shaft oper
ating at constant speed without overloads. For this type of stress cycle
the maximum and minimum stresses are equal. In keeping with the
conventions established in Chap. 2 the minimum stress is the lowest
Fig. 122. Typical fatigue stress cycles,
(c) irregular or random stress cycle.
(a) Reversed stress; (6) repeated stress;
algebraic stress in the cycle. Tensile stress is considered positive, and
compressive stress is negative. Figure 1226 illustrates a repeated stress
cycle in which the mp^ximum stress omax and minimum stress a^nin are not
equal. In this illustration they are both tension, but a repeated stress
cycle could just as well contain maximum and minimum stresses of
opposite signs or both in compression. Figure 122c illustrates a com
plicated stress cycle which might be encountered in a part such as an
aircraft wing which is subjected to periodic unpredictable overloads due
to gusts.
A fluctuating stress cycle can be considered to be made up of two com
ponents, a mean, or steady, stress (Xm, and an alternating, or variable,
1 Common types of fatigue machines are described in the references listed, at the
end of this chapter and in the Manual on Fatigue Testing, ASTM Spec. Tech. Publ.
91, 1949.
Sec. 123]
Fatigue of Metals 299
stress (Xa. We must also consider the range of stress Cr. As can be seen
from Fig. 1226, the range of stress is the algebraic difference between the
maximum and minimum stress in a cycle.
^T O'max O^n
The alternating stress, then, is onehalf the range of stress.
(7„  2
(121)
(122)
The mean stress is the algebraic mean of the maximum and minimum
stress in the cycle.
<Tm — ^ (126)
Another quantity which is sometimes used in presenting fatigue data is
the stress ratio R. Stress ratio is defined as
R =
(124)
123. The >SA^ Curve
The basic method of presenting engineering fatigue data is by means
of the SN curve, which represents the dependence of the life of the
specimen, in number of cycles to failure, A'^, on the maximum applied
stress 0. Most investigations of the fatigue properties of metals have
been made by means of the rotatingbeam machine, where the mean stress
is zero. Figure 123 gives typical SN curves for this type of test. Cases
60
t/i
8 50
o
!^'40
(/)
1^30
^20
"O
O
o
o
V.
^^/V
^
d Sf6
^el
,— .
limit
^^^
.^_^AIum
'nam
alloy
\Q~
10^
10^
10^
10
Number of cycles to failure, N
Fi9. 123. Typical fatigue curves for ferrous and nonferrous metals.
300 Applications to Materials Testing [Chap. 1 2
where the mean stress is not zero are of considerable practical interest.
These will be discussed later in the chapter.
As can be seen from Fig. 123, the number of cj^cles of stress which a
metal can endure before failure increases with decreasing stress. Unless
otherwise indicated, N is taken as the number of cycles of stress to cause
complete fracture of the specimen. This is made up of the number of
cycles to initiate a crack and the number of cycles to propagate the crack
completely through the specimen. Usually no distinction is made
between these two factors, although it can be appreciated that the
number of cycles for crack propagation will vary with the dimensions
of the specimen. Fatigue tests at low stresses are usually carried out
for 10^ cycles and sometimes to 5 X 10^ cycles for nonferrous metals.
For a few important engineering materials such as steel and titanium,
the SN curve becomes horizontal at a certain limiting stress. Below
this limiting stress, which is called the fatigue limit, or endurance limit,
the material can presumably endure an infinite number of cycles without
failure. Most nonferrous metals, like aluminum, magnesium, and copper
alloys, have an SN curve which slopes gradually downward with increas
ing number of cycles. These materials do not have a true fatigue limit
because the SN curve never becomes horizontal. In such cases it is
common practice to characterize the fatigue properties of the material
by giving the fatigue strength at an arbitrary number of cycles, for
example, 10* cycles. The reasons why certain materials have a fatigue
limit are not known, although a hypothesis regarding this impor
tant question will be discussed later in the chapter.
The usual procedure for determining an SN curve is to test the first
specimen at a high stress where failure is expected in a fairly short num
ber of cycles, e.g., at about twothirds the static tensile strength of the
material. The test stress is decreased for each succeeding specimen until
one or two specimens do not fail in the specified number of cycles, which
is usually at least 10'^ cycles. The highest stress at which a runout (non
failure) is obtained is taken as the fatigue limit. For materials without
a fatigue limit the test is usually terminated for practical considerations
at a low stress where the life is about 10* or 5 X 10* cycles. The SN
curve is usually determined with about 8 to 12 specimens. It will gener
ally be found that there is a considerable amount of scatter in the results,
although a smooth curve can usually be drawn through the points with
out too much difficulty. However, if several specimens are tested at the
same stress, there is a great amount of scatter in the observed values of
number of cycles to failure, frequently as much as one log cycle between
the minimum and maximum value. Further, it has been shown ^ that
the fatigue limit of steel is subject to considerable variation and that a
1 J. T. Ransom and R. F. Mehl, Trans. AIMS, vol. 185, pp. 364365, 1949.
Sec. 124] Fatisuc of Metals 301
fatigue limit determined in the manner just described can be considerably
in error. The statistical nature of fatigue will be discussed in the next
section.
An interesting test for obtaining a more rapid estimate of the fatigue
limit than is possible by conventional means was proposed by Prot.^ In
this method, each specimen is started at an initial stress below the
expected value of the fatigue limit, and the stress is progressively
increased at a constant rate until fracture occurs. Several specimens
are tested at different values of stress increase per cycle. Prot suggested
that a linear relationship should exist between the stress at which frac
ture occurs and y/ a, w^here a is the stress increase per cycle. The fatigue
limit is obtained from this plot by extrapolation to y/ a = 0. Profs
method has undergone considerable investigation and modification' and
appears useful for the rapid determination of the fatigue limit of ferrous
materials.
A modification of the Prot method is sometimes used w^hen a special
machine equipped to provide a continuously increasing stress is not avail
able or when the number of specimens is not large. The initial stress
level is taken at about 70 per cent of the estimated fatigue limit. The
test is run for a fixed number of cycles, for example, 10^, and if failure
does not occur, the stress is raised by a certain amount. Another unit
of cycles is applied at this stress, and the process is continued until failure
occurs. The fatigue limit of the specimen is taken as the stress halfway
between the breaking stress and the highest stress at which the specimen
survived. Results obtained by this step method and the Prot method
may not produce values of fatigue limit in agreement with those obtained
from testing at constant stress, because of changes which can occur in
the metal during testing at stresses below the fatigue limit. For example,
certain metals can be strengthened b}^ "coaxing" at stresses below the
fatigue limit. This topic is discussed in greater detail in Sec. 1213.
124. Statistical Nature of Fatigue
A considerable amount of interest has been shown in the statistical
analysis of fatigue data and in the reasons for the variability in fatigue
test results. A more complete description of the statistical techniques
will be given in Chap. 16. However, it is important here to gain an
acquaintance with the concept of the statistical approach so that existing
fatigue data can be properly evaluated. Since fatigue life and fatigue
limit are statistical quantities, it must be realized that considerable devi
1 M. Prot, Rev. mH., vol. 34, p. 440, 1937.
2 H. T. Corten, T. Dimoff, and T. J. Dolan, Proc. ASTM, vol. 54, pp. 875902,
1954.
302 Applications to Materials Testing
[Chap. 12
ation from an average curve determined with only a few specimens is to
be expected. It is necessary to think in terms of the probabihty of a
specimen attaining a certain Hfe at a given stress or the probability of
failure at a given stress in the vicinity of the fatigue limit. To do this
requires the testing of considerably more specimens than in the past so
that the statistical parameters^ for estimating these probabilities can be
determined. The basic method for expressing fatigue data should then
be a threedimensional surface representing the relationship between
.i
V\\^ /i verage curve P=0.50
\ \ 1 NN^
\ \ \
\ \/
\l \
\
\. V"" — P=0.99
\ P = 0.90
P=0.99
P=0.01
H
U.IU
) 1  U.jU
P = O.OJ
\
1
1
1
Mean fc
itfgue
limit
1
/V, /Vp
^n
Number of cycles (log scale)
Fig. 1 24. Representation of fatigue data on a probability basis.
stress, number of cycles to failure, and probability of failure. Figure
124 shows how this can be presented in a twodimensional plot.
A distribution of fatigue life at constant stress is illustrated schemati
cally in this figure, and based on this, curves of constant probability of
failure are drawn. Thus, at ci, 1 per cent of the specimens would be
expected to fail at Ni cycles, 50 per cent at A''2 cycles, etc. The figure
indicates a decreasing scatter in fatigue life with increasing stress, which
is usually found to be the case. The statistical distribution function
which describes the distribution of fatigue life at constant stress is not
accurately known, for this would require the testing of over 1,000 identi
cal specimens under identical conditions at a constant stress. Muller
Stock tested 200 steel specimens at a single stress and found that the
^ The chief statistical parameters to be considered are the estimate of the mean
(average) and standard deviation (measure of scatter) of the population.
2 H. MullerStock, Mitteilung Kohle u. Eisenforsch. G. m. b. H., vol. 8, pp. 83107,
1938.
Sec. 124] Fatigue oF Metals 303
frequency distribution of N followed the Gaussian, or normal, distribu
tion if the fatigue life was expressed as log N. For engineering purposes
it is sufficiently accurate to assume a logarithmic normal distribution of
fatigue life at constant life in the region of the probability of failure of
P = 0.10 to P = 0.90. However, it is frequently important to be able
to predict the fatigue life corresponding to a probability of failure of
1 per cent or less. At this extreme limit of the distribution the assump
tion of a lognormal distribution of life is no longer justified, although it
is frequently used. Alternative approaches have been the use of the
extremevalue distribution ^ or Weibull's distribution. 
For the statistical interpretation of the fatigue limit we are concerned
with the distribution of stress at a constant fatigue life. The fatigue
limit of steel was formerly considered to be a sharp threshold value,
below which all specimens w^ould presumably have infinite lives. How
ever, it is now recognized that the fatigue limit is really a statistical
quantity which requires special techniques for an accurate determination.
For example, in a heattreated alloy forging steel the stress range which
would include the fatigue limits of 95 per cent of the specimens could
easily be from 40,000 to 52,000 psi. An example of the errors which can
be introduced by ordinary testing with a few specimens is illustrated in
Fig. 125. This figure summarizes^ ten SN curves determined in the
conventional manner for the same bar of alloy steel, each curve being
based on ten specimens. The specimens were as identical as it was possi
ble to make them, and there was no excessive scatter or uncertainty as to
how to draw the SN curves. Yet, as can be seen from the figure, there is
considerable difference in the measured values of the fatigue limit for the
steel due to the fact that the curves were based on insufficient data.
In determining the fatigue limit of a material, it should be recognized
that each specimen has its own fatigue limit, a stress above which it will
fail but below which it will not fail, and that this critical stress varies
from specimen to specimen for very obscure reasons. It is known that
inclusions in steel have an important effect on the fatigue limit and its
variability, but even vacuummelted steel shows appreciable scatter in
fatigue limit. The statistical problem of accurately determining the
fatigue limit is complicated by the fact that we cannot measure the indi
vidual value of the fatigue limit for any given specimen. We can only
test a specimen at a particular stress, and if the specimen fails, then the
stress was somewhere above the fatigue limit of the specimen. Since
the specimen cannot be retested, even if it did not fail at the test stress,
1 A. M. Freudenthal and E. J. Gumbel, J. Am. Statist. Assoc, vol. 49, pp. 575597,
1954.
2 W. WeibuU, J. Appl. Mech., vol. 18, no. 3, pp. 293297, 1951.
3 J. T. Ransom, discussion in ASTM Spec. Tech. Puhl. 121, pp. 5963, 1952.
304 Applications to Materials Testins
[Chap. 12
we have to estimate the statistics of the fatigue limit by testing groups of
specimens at several stresses to see how many fail at each stress. Thus,
near the fatigue limit fatigue is a "gono go" proposition, and all that we
can do is to estimate the behavior of a universe of specimens by means
of a suitable sample. The two statistical methods which are used for
making a statistical estimate of the fatigue limit are called probit analysis
3,000
70,000
60,000
6" 2 !0 9175384
50,000
40,000
Cycles to failure
Fis. 1 25. Summary of B'N curves, each based on 10 specimens, drawn from the same
bar of steel. (J. T. Ransom, ASTM Spec. Tech. Publ. 121, p. 61, 1952.)
and the staircase method. The procedures for applying these methods of
analysis to the determination of the fatigue hmit will be given in Chap. 16.
125. Structural Features of Fatigue
Only a small fraction of the effort devoted to fatigue research has been
concerned with the study of the basic structural changes that occur in a
metal when it is subjected to cyclic stress. Fatigue has certain things in
Sec. 125] Fatigue of Metals 305
common with plastic flow and fracture under static or unidirectional
deformation. The work of Gough^ has shown that a metal deforms
under cyclic strain by slip on the same atomic planes and in the same
crystallographic directions as in unidirectional strain. Whereas with
unidirectional deformation slip is usually widespread throughout all the
grains, in fatigue some grains will show slip lines while other grains will
give no evidence of slip. Slip lines are generally formed during the first
few thousand cycles of stress. Successive cycles produce additional slip
bands, but the number of slip bands is not directly proportional to the
number of cycles of stress. In many metals the increase in visible slip
soon reaches a saturation value, which is observed as distorted regions
of heavy slip. Cracks are usually found to occur in the regions of heavy
deformation parallel to what was originally a slip band. Slip bands have
been observed at stresses below the fatigue limit of ferrous materials.
Therefore, the occurrence of slip during fatigue does not in itself mean
that a crack will form.
A study of crack formation in fatigue can be facilitated by interrupting
the fatigue test to remove the deformed surface by electropolishing.
There will generally be several slip bands which are more persistent than
the rest and which will remain visible when the other slip lines have been
polished away. Such slip bands have been observed after only 5 per
cent of the total life of the specimen.^ These persistent slip bands are
embryonic fatigue cracks, since they open into wide cracks on the appli
cation of small tensile strains. Once formed, fatigue cracks tend to
propagate initially along slip planes, although they may later take a
direction normal to the maximum applied tensile stress. Fatiguecrack
propagation is ordinarily transgranular.
An important structural feature which appears to be unique to fatigue
deformation is the formation on the surface of ridges and grooves called
slipband extrusions and slipband intrusions J Extremely careful metal
lography on tapered sections through the surface of the specimen has
shown that fatigue cracks initiate at intrusions and extrusions.'* There
fore, these structural features are the origin of the persistent slip bands,
or fissures, discussed in the previous paragraph. The study of slipband
intrusions and extrusions has been undertaken too recently to uncover
all the factors responsible for their formation. However, it appears that
intrusions and extrusions are produced at local soft spots in the crystal,
and this suggests that cross slip is needed for their formation. This
1 H. J. Gough, Proc. ASTM, vol. 33, pt. 2, pp. 3114, 1933.
2 G. C. Smith, Proc. Roy. Soc. (London), vol. 242A, pp. 189196, 1957.
3 P. J. E. Forsyth and C. A. Stubbington, J. Inst. Metals, vol. 83, p. 395, 19551956.
* W. A. Wood, Some Basic Studies of Fatigue in Metals, in "Fracture," John
Wiley & Sons, Inc., New York, 1959.
306 Applications to Materials Testing [Chap. 1 2
hypothesis is borne out by the fact that fatigue failure is difficult to pro
duce in certain ionic crystals which do not easily undergo cross slip and by
the fact that it is not possible to produce fatigue failure in zinc crystals,
which are oriented to deform only in easy glide.
In considering the structural changes produced by fatigue, it is advisa
ble to differentiate between tests conducted at high stresses or strain
amplitudes, where failure occurs in less than about 10^ cycles of stress,
and tests carried out at low stresses, where failure occurs in more than
10^ cycles. Structural features produced in the highstress region of the
SN curve bear a strong resemblance to those produced by unidirec
tional deformation. An annealed metal usually undergoes moderate
strain hardening with increasing cycles in the highstress region. Coarse
slip bands are formed, and there is appreciable asterism in the Xray
diffraction pattern. However, in the lowstress region slip lines are very
fine and are dif&cult to distinguish by ordinary metallographic techniques.
There is essentially no strain hardening or distortion in the Xray diffrac
tion pattern. For copper specimens tested in the highstress region, the
stored energy is released over a fairly narrow temperature range during
annealing. This represents energy release due to both recovery and
recrystallization, just as would be expected for a metal plastically
deformed in tension. When the copper is fatigued in the lowstress
region, the stored energy is released over a wide range of temperature,
as would occur if only recovery took place. ^
A study of the dislocation structure in thin films of aluminum^ has
shown that for high fatigue stresses dislocation networks are formed simi
lar to those formed on unidirectional loading. At low fatigue stresses
the metal contains a high density of dislocation loops similar to those
found in quenched specimens. This is a good indication that large num
bers of point defects are produced during fatigue.
There are a number of other indications that cyclic deformation results
in a higher concentration of vacancies than cold working by unidirec
tional deformation. The difference in the release of stored energy
between fatigued and coldworked copper is in line with what would be
expected from a large concentration of point defects. The fact that
initially coldworked copper becomes softer as a result of fatigue^ can be
explained by the generation of point defects which allow the metal partly
to recover by permitting dislocations to climb out of the slip plane.
Agehardening aluminum alloys in the precipitationhardened condition
can be overaged by fatigue deformation at room temperature. This sug
1 L. M. Clarebrough, M. E. Hargreaves, G. W. West, and A. K. Head, Proc. Roy.
Soc. (London), vol. 242A, pp. 160166, 1957.
2 R. L. Segall and P. G. Partridge, Phil. Mag., vol. 4, pp. 912919, 1959.
3 N. H. Polakowski and A. Palchoudhuri, Proc. ASTM, vol. 54, p. 701, 1954.
Sec. 126] Fatigue of Metals 307
gests that vacancies produced by fatigue are available to accomplish the
diffusion required for the overaging process.^ Moreover, the fatigue
strength increases markedly on going from 20 to — 190°C, where vacancy
movement is negligible. However, the fact that fatigue fracture can be
produced at 4°K indicates that a temperatureactivated process such as
the diffusion of vacancies is not essential for fatigue failure. 
The process of the formation of a fatigue crack is often divided into
three stages.' The primary stage occurs only in metals where the applied
stress level is above the initial static yield stress. Widespread bulk defor
mation occurs until the metal strain hardens to the point where it can
withstand the applied stress. Depending upon the stress, the first stage
will last for 10^ to 10* cycles. The second stage comprises the major part
of the fatigue life of a specimen. It extends from the initial widespread
strain hardening to the formation of a visible crack. During the second
stage of fatigue the crack is initiated. The third stage of fatigue con
sists of the propagation of the crack to a size large enough to cause failure.
There is considerable evidence that a fatigue crack is formed before
about 10 per cent of the total life of the specimen has elapsed, although
the crack cannot be readily detected, except by repeated electropolishing,
until many cycles later. The principal evidence for this'*'^ is that anneal
ing after only a small fraction of the expected total fatigue life does not
significantly increase the fatigue life. It has been concluded that the
damage produced by this small number of cycles must be in the nature
of a crack.
126. Theories of Fatigue
It is perhaps unnecessary to state that no mechanism or theory has
been proposed which adequately explains the phenomenon of fatigue.
For one thing, it is unlikely that our knowledge of the structural changes
produced by fatigue is at all complete. Many of the theories that exist
have been qualitative and base their acceptance mainly on the fact that
the analysis yields a stresslog A'' relationship similar to the observed
SN curve. However, this may not necessarily be a satisfactory criterion,
for many assumed mechanisms can lead to a prediction of the general
shape of the fatigue curve.
1 T. Broom, J. H. Molineux, and V. N. Whittaker, J. Inst. Metals, vol. 84, pp. 357
363, 195556.
2 R. D. McCammon and H. M. Rosenberg, Proc. Roy. Soc. (London), vol. 242A,
p. 203, 1957.
3 A. K. Head, /. Mech. and Phys. Solids, vol. 1, pp. 134141, 1953.
* G. M. Sinclair and T. J. Dolan, Proc. First Natl. Congr. Appl. Mech., 1951, pp.
647651.
^ N. Thompson, N. Wadsworth, and N. Louat, Phil. Mag., vol. 1, pp. 113126, 1956.
308 Applications to Materials Testing [Chap. 1 2
Orowan's Theory
Orowan's theory of fatigue^ was one of the earUest generally accepted
explanations for the fatigue process. This theory leads to the prediction
of the general shape of the SN curve, but it does not depend on any
specific deformation mechanism other than the concept that fatigue defor
mation is heterogeneous. The metal is considered to contain small, weak
regions, which may be areas of favorable orientation for slip or areas of
high stress concentration due to metallurgical notches such as inclusions.
It was assumed that these small regions could be treated as plastic regions
in an elastic matrix. Orowan showed that for repeated cycles of constant
stress amplitude the plastic regions will experience an increase in stress
and a decrease in strain as the result of progressive localized strain harden
ing. He further showed that the total plastic strain (sum of positive and
negative strains) converges toward a finite value as the number of cycles
increases toward infinity. This limiting value of total plastic strain
increases with an increase in the stress applied to the specimen. The
existence of a fatigue limit hinges upon the fact that below a certain
stress the total plastic strain cannot reach the critical value required for
fracture. However, if the stress is such that the total plastic strain in
the weak region exceeds the critical value, a crack is formed. The crack
creates a stress concentration, and this forms a new localized plastic
region in which the process is repeated. This process is repeated over
and over until the crack becomes large enough so that fracture occurs on
the application of the full tensile stress of the cycle. The essence of this
theory is that localized strain hardening uses up the plasticity of the
metal so that fracture takes place.
Wood^s Concept of Fatigue
W. A. Wood,^ who has made many basic contributions to the under
standing of the mechanism of fatigue, has evolved a concept of fatigue
failure which does not require localized strain hardening for fatigue defor
mation to occur. He interprets microscopic observations of slip produced
by fatigue as indicating that the slip bands are the result of a systematic
buildup of fine slip movements, corresponding to movements of the order
of 10~^ cm rather than steps of 10"^ to 10~* cm, which are observed for
static slip bands. Such a mechanism is believed to allow for the accom
modation of the large total strain (summation of the microstrain in each
cycle) without causing appreciable strain hardening. Figure 126 illus
trates Wood's concept of how continued deformation by fine slip might
lead to a fatigue crack. The figures illustrate schematically the fine
1 E. Orowan, Proc. Roy. Soc. {London), vol. 171A, pp. 79106, 1939.
2 W. A. Wood, Bull. Inst. Metals, vol. .3, pp. 56, September, 1955.
Sec. 1 26]
Fatigue of Metals 309
structure of a slip band at magnifications obtainable with the electron
microscope. Slip produced by static deformation would produce a con
tour at the metal surface similar to that shown in Fig. 126a. In con
trast, the backandforth fine slip movements of fatigue could build up
notches (Fig. 1266) or ridges (Fig. 126c) at the surface. The notch
would be a stress raiser with a notch root of atomic dimensions. Such a
[b]
[c)
Fig. 126. W. A. Wood's concept of microdeformation leading to formation of fatigue
crack, (a) Static deformation; (6) fatigue deformation leading to surface notch
(intrusion) ; (c) fatigue deformation leading to slipband extrusion.
situation might well be the start of a fatigue crack. This mechanism
for the initiation of a fatigue crack is in agreement wdth the facts that
fatigue cracks start at surfaces and that cracks have been found to initiate
at slipband intrusions and extrusions.
Dislocation Models for Fatigue
The growing awareness of the role played by subtle changes in surface
topography in initiating fatigue cracks has led to several dislocation
models for the generation of slipband intrusions and extrusions. Cottrell
and HulP have suggested a model involving the interaction of edge dis
locations on two slip systems, while Mott^ has suggested one involving
the cross slip of screw dislocations. Fatigue experiments on ionic crys
tals^ tend to support the Mott mechanism and to disprove the Cottrell
Hull model.
Theory of the Fatigue Limit
One of the puzzling questions in fatigue is why certain metals exhibit
an SN curve with a welldefined fatigue limit, while other metals do not
have a fatigue limit. The answer to this question appears to have been
1 A. H. Cottrell and D. Hull, Proc. Roy. Soc. (London), vol. 242A, pp. 211213, 1957.
2 N. F. Mott, Acta Met., vol. 6, pp. 195197, 1958; see also A. J. Kennedy, Phil.
Mag., ser. 8, vol. 6, pp. 4953, 1961.
3 A. J. McEvily, Jr., and E. S. Machlin, Critical Experiments on the Nature of
Fatigue in Crystalline Materials, in "Fracture," John Wiley & Sons, Inc., New York,
1959.
310 Applications to Materials Testing [Chap. 12
given by Rall}^ and Sinclair/ who noted that metals which undergo strain
aging have an SN curve with a sharp knee and a welldefined fatigue
limit. Their tests with mild steel showed that as the total carbon and
nitrogen content was decreased, so that the tendency for strain aging
decreased, the SN curve flattened out and the knee occurred at a larger
number of cycles than if the carbon content were higher. Similar results
were found by Lipsitt and Home." They proposed that the fatigue limit
represents the stress at which a balance occurs between fatigue damage
and localized strengthening due to strain aging. The correlation is fairly
good between materials which show both strain aging and a fatigue limit.
Lowcarbon steel, titanium, molybdenum, and aluminum7 per cent
magnesium^ alloy are good examples. Heattreated steel exhibits a
definite fatigue limit; yet it does not ordinarily show strain aging in the
tension test. However, only very localized strain aging is required to
affect fatigue properties, and it is quite likely that the fatigue test is
more sensitive to strain aging than the tension test.
1 27. Effect of Stress Concentration on Fatigue
Fatigue strength is seriously reduced by the introduction of a stress
raiser such as a notch or hole. Since actual machine elements invariably
contain stress raisers like fillets, keyways, screw threads, press fits, and
holes, it is not surprising to find that fatigue cracks in structural parts
usually start at such geometrical irregularities. One of the best ways of
minimizing fatigue failure is by the reduction of avoidable stress raisers
through careful design^ and the prevention of accidental stress raisers by
careful machining and fabrication. While this section is concerned with
stress concentrations resulting from geometrical discontinuities, stress
concentration can also arise from surface roughness and metallurgical
stress raisers such as porosity, inclusions, local overheating in grinding,
and decarburization. These factors will be considered in other sections
of this chapter.
The effect of stress raisers on fatigue is generally studied by testing
specimens containing a notch, usually a V notch or a circular notch. It
has been shown in Chap. 7 that the presence of a notch in a specimen
1 F. C. Rally and G. M. Sinclair, "Influence of Strain Aging on the Shape of the
SN Diagram," Department of Theoretical and Applied Mechanics, University of
Illinois, Urbana, 111., 1955; see also J. C. Levy and S. L. Kanitkar, J. Iron Steel Inst.
(London), vol. 197, pp. 296300, 1961.
2 H. A. Lipsitt and G. T. Home, Proc. ASTM, vol. 57, pp. 587600, 1957.
^ Broom, Molineux, and Whittaker, op. cit.
* For examples of good design practice, see J. S. Caswell, Prod. Eng., January, 1947,
pp. 118119.
)ec.
127] Fatisue of Metals 311
under uniaxial load introduces three effects: (1) there is an increase or
concentration of stress at the root of the notch; (2) a stress gradient is
set up from the root of the notch in toward the center of the specimen;
(3) a triaxial state of stress is produced.
The ratio of the maximum stress to the nominal stress is the theoretical
stressconcentration factor Ki. As was discussed in Sec. 213, values of
Kt can be computed from the theory of elasticity for simple geometries
and can be determined from photoelastic measurements for more com
plicated situations. Most of the available data on stressconcentration
factors have been collected by Peterson. ^ It is often desirable to include
the effect of the biaxial state of stress at the root of a notch in the value
of the stressconcentration factor. The distortionenergy criterion of
yielding for biaxial stress can be expressed by
a, = (7i(l  C + C2)^^ (125)
where C = crz/ai and ao = 0. If we divide both sides of Eq. (125) by
the nominal stress, we get the expression
Kt' = Kt{l  C + Cy^ (126)
where Kt' is the stress concentration factor including both combined stress
and stress concentration.
The effect of notches on fatigue strength is determined by comparing
the SN curves of notched and unnotched specimens. The data for
notched specimens are usually plotted in terms of nominal stress based
on the net section of the specimen. The effectiveness of the notch in
decreasing the fatigue limit is expressed by the fatiguestrength reduction
factor, or fatiguenotch factor, Kf. This factor is simply the ratio of the
fatigue limit of unnotched specimens to the fatigue limit of notched speci
mens. For materials which do not exhibit a fatigue limit the fatigue
notch factor is based on the fatigue strength at a specified number of
cycles. Values of Kf have been found to vary with (1) severity of the
notch, (2) the type of notch, (3) the material, (4) the type of loading, and
(5) the stress level. The values of Kf published in the literature are
subject to considerable scatter and should be carefully examined for their
limitations and restrictions. However, two general trends are usually
observed for test conditions of completely reversed loading. First,
Kf is usually less than Kt, and, second, the ratio of Kf/Kt decreases
as Kt increases. Thus, very sharp notches (high Kt) have less effect on
fatigue strength than would be expected from their high value of Kt.
This is in agreement with observations that fatigue cracks can exist in a
1 R. E. Peterson, "Stressconcentration Design Factors," John Wiley & Sons, Inc.,
New York, 195:1
312 Applications to Materials Testing
[Chap. 12
specimen for millions of cycles without propagating. ^ However, this
should not be interpreted as license to assume that a sharp notch or crack
can be tolerated in a structure.
The notch sensitivity of a material in fatigue is usually expressed by a
notchsensitivity index q.
Q =
Ki
1
Kt  1
or
Kt  1
Kt'
1
(127)
where q = notchsensitivity index
Kf — notchfatigue factor
= fatigue limit unnotched/fatigue limit notched
Kt = theoretical stressconcentration factor = omax/cnom
Kt' = theoretical factor which combines Kt and a biaxial stress factor
Equation (127) was chosen so that a material which experiences no
reduction in fatigue strength due to a notch has an index of g = 0, while
a material in which the notch has its full theoretical effect has a notch
sensitivity index of g = 1. However, q is not a true material constant
since it varies with the severity and type of notch, the size of the speci
men, and the type of loading. The notch sensitivity increases with sec
tion size and tensile strength. Thus, because of increased q it is possible
under certain circumstances to decrease the fatigue performance of a
member by increasing the hardness or tensile strength.
Fig. 1 27. Stress distribution at a notch in bending.
The stress gradient, or slope of the stressdistribution curve near the
root of the notch, has an important influence on the notch sensitivity.
Figure 127 illustrates the stress distribution in a notched bar in bending.
The maximum stress produced by the notch is omax, and the nominal stress,
neglecting the notch, is o„. The unnotched fatigue limit of the material
is (Tg. This stress is reached at a depth 5 below the root of the notch.
» N. E. Frost, Engineer, vol. 200, pp. 464, 501, 1955.
Sec. 127] Fatigue of Metals 313
The stress gradient can then be written
dy 5
This expression can also be written in terms of the notch radius r.
HO" ^ O^inax
dy r
Combining these two expressions and assuming that faikire occurs when
in
1 1
0.8
^
rs
, — "
^
1
'
/
^i
\.
^^
■^0.6
/ /
/ \
.^\
//
X
\. ^^ Quenched and tempered steel
i ^^
\h
/
\^
^^Annealed and normalized steel
\
Aluminum alloy
1
^ 0.2
2
I
f
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
r, in.
Fig. 128. Variation of notchsensitivity index with notch radius for materials of
different tensile strength. {R. E. Peterson, in G. Sines and J. L. Waisman {eds.),
"Metal Fatigue," p. 301, McGrawHill Book Company, Inc., New York, 1959.)
the stress ae at the depth 5 equals the fatigue strength K/Cn results in
Kf = /vYl  ^^ (128)
Typical values^ of Ci are 2.5 for bending and axial loading and 1.2 for
torsion. If Eq. (128) is substituted into Eq. (127) and Kt is replaced
by Kt', we get a relationship between q and notch radius.'^
q= 1
(129)
where a = Cih[Kt'/{Kt'  1)]. Figure 128 shows typical values of
notchsensitivity index plotted against notch radius. Note the effect
of the strength of the material (see Prob. 12.1). Since the stress gradient
will vary with section size in the same way as notch radius, q will also
1 M. M. Leven, Proc. Soc. Exptl. Stress Anal., vol. 13, no. 1, p. 207, 1955.
2 R. E. Peterson, Notchsensitivity, in G. Sines and J. L. Waisman (eds.), "Metal
Fatigue," McGrawHill Book Company, Inc., New York, 1959.
314 Applications to Materials Testing [Chap. 12
increase with specimen diameter. In addition, there is a measurable
effect of grain size on notchsensitivity index. Fine grain size results in
a higher q than coarse grain size.
Several hypotheses have been made to explain the variation of notch
sensitivity with notch radius, section size, and grain size. One hypothe
sis assumes that failure is determined by the volume of material that is
stressed to within a small percentage, say, 5 per cent, of the maximum
stress. This involves a statistical argument that the probability of find
ing a flaw or critical crack nucleus increases with the volume of highly
stressed material. Another viewpoint is that the stress gradient across
a grain is the critical factor. For a fine grain size the stress gradient is
low, and the value of q is large. Geometrically similar notches will not
produce the same stress gradient across the grains if the grain size is
equal in differentdiameter specimens. The specimen with the larger
diameter will have the lower stress gradient across a grain.
1 28. Size Effect
An important practical problem is the prediction of the fatigue per
formance of large machine members from the results of laboratory tests
on small specimens. Experience has shown that in most cases a size effect
exists; i.e., the fatigue strength of large members is lower than that of
small specimens. A precise study of this effect is difficult for several
reasons. It is extremely difficult, if not altogether impossible, to prepare
geometrically similar specimens of increasing diameter which have the
same metallurgical structure and residual stress distribution throughout
the cross section. The problems in fatigue testing largesized specimens
are considerable, and there are few fatigue machines which can accom
modate specimens having a wide range of cross sections.
Changing the size of a fatigue specimen usually results in a variation
in two factors. First, increasing the diameter increases the volume or
surface area of the specimen. The change in amount of surface is of sig
nificance, since fatigue failures usually start at the surface. Second, for
plain or notched specimens loaded in bending or torsion, an increase in
diameter usually decreases the stress gradient across the diameter and
increases the volume of material which is highly stressed.
Experimental data on the size effect in fatigue are contradictory and
not very complete. For tests in reversed bending and torsion, some
investigators have found no change in fatigue limit with specimen diam
eter, while more commonly it is observed that the fatigue limit decreases
with increasing diameter. For mild steel the decrease in bending fatigue
limit for diameters ranging from 0.1 to 2 in. does not exceed about 10 per
Sec. 129] Fatigue of Metals 315
cent. Horger's data' for steel shafts tested in reversed bending (Table
121) show that the fatigue limit can be appreciably reduced in large
section sizes.
Table 121
Fatigue Limit of Normalized Plaincarbon Steel
IN Reversed Bending
Specimen diam,
Fatigue limit,
in.
psi
0.30
36,000
1.50
29,000
6.00
21,000
No size effect has been found for smooth plaincarbonsteel fatigue
specimens with diameters ranging from 0.2 to 1.4 in. when tested in axial
tensioncompression loading. However, when a notch is introduced into
the specimen, so that a stress gradient is produced, a definite size effect
is observed. These important experiments support the idea that a size
effect in fatigue is due to the existence of a stress gradient. The fact
that large specimens with shallow stress gradients have lower fatigue
limits is consistent with the idea that a critical value of stress must be
exceeded over a certain finite depth of material for failure to occur. This
appears to be a more realistic criterion of size effect than simply the ratio
of the change in surface area to the change in specimen diameter. The
importance of stress gradients in size effect helps explain why correlation
between laboratory results and service failure is often rather poor.
Actual failures in large parts are usually directly attributable to stress
concentrations, either intentional or accidental, and it is usually impossi
ble to duplicate the same stress concentration and stress gradient in a
smallsized laboratory specimen.
1 29. Surface Effects and Fatigue
Practically all fatigue failures start at the surface. For many common
types of loading, like bending and torsion, the maximum stress occurs
at the surface so that it is logical that failure should start there. How
ever, in axial loading the fatigue failure nearly always begins at the
surface. There is ample evidence that fatigue properties are very sensi
tive to surface condition. The factors which affect the surface of a
fatigue specimen can be divided roughly into three categories, (1) surface
1 O. J. Horger, Fatigue Characteristics of Large Sections, in "Fatigue," American
Society for Metals, Metals Park, Ohio, 1953.
2 C. E. PhiUips and R. B. Heywood, Proc. Inst. Mech. Engrs. (London), vol. 165,
pp. 113124, 1951.
316 Applications to Materials Testing
[Chap. 12
roughness or stress raisers at the surface, (2) changes in the fatigue
strength of the surface metal, and (3) changes in the residual stress con
dition of the surface. In addition, the surface is subjected to oxidation
and corrosion.
Surface Roughness
Since the early days of fatigue investigations, it has been recognized
that different surface finishes produced by different machining procedures
can appreciably affect fatigue performance. Smoothly polished speci
mens, in which the fine scratches (stress raisers) are oriented parallel with
the direction of the principal tensile stress, give the highest values in
fatigue tests. Such carefully polished specimens are usually used in
Table 122
Fatigue Life of SAE 3130 Steel Specimens Tested under
Completely Reversed Stress at 95,000 PSI^^
Type of finish
Surface roughness, lAn.
Median fatigue life, cycles
Latheformed
105
6
5
7
2
7
24,000
Partly handpolished
Handpolished
Ground
91,000
137,000
217,000
Ground and polished
Superfinished
234,000
212,000
t p. G. Fluck, Proc. ASTM, vol. 51, pp. 584592, 1951.
laboratory fatigue tests and are known as "par bars." Table 122 indi
cates how the fatigue life of cantileverbeam specimens varies with the
type of surface preparation. Extensive data on this subject have been
published by Siebel and Gaier.^
Changes in Surface Properties
Since fatigue failure is so dependent on the condition of the surface,
anything that changes the fatigue strength of the surface material will
greatly alter the fatigue properties. Decarburization of the surface of
heattreated steel is particularly detrimental to fatigue performance.
Similarly, the fatigue strength of aluminumalloy sheet is reduced when
a soft aluminum coating is applied to the stronger agehardenable alumi
numalloy sheet. Marked improvements in fatigue properties can result
from the formation of harder and stronger surfaces on steel parts by
1 E. Siebel and M. Gaier, VDI Zt., vol. 98, pp. 17151723, 1956; abstracted in Engi
neer's Digest, vol. 18, pp. 109112, 1957.
Sec. 129] Fatigue of Metals 317
carburizing and nitriding. ' However, since favorable compressive resid
ual stresses are produced in the surface by these processes, it cannot be
considered that the higher fatigue properties are due exclusively to the
formation of higherstrength material on the surface. The effectiveness
of carburizing and nitriding in improving fatigue performance is greater
for cases where a high stress gradient exists, as in bending or torsion,
than in an axial fatigue test. The greatest percentage increase in fatigue
performance is found when notched fatigue specimens are nitrided. The
amount of strengthening depends on the diameter of the part and the
depth of surface hardening. Improvements in fatigue properties similar
to those caused by carburizing and nitriding may also be produced by
flame hardening and induction hardening. It is a general characteristic
of fatigue in surfacehardened parts that the failure initiates at the inter
face between the hard case and the softer case, rather than at the surface.
Electroplating of the surface generally decreases the fatigue limit of
steel. Chromium plating is particularly difficult to accomplish without
impairment of fatigue properties, while a softer cadmium plating is
believed to have little effect on fatigue strength. The particular plating
conditions used to produce an electroplated surface can have an appreci
able effect on the fatigue properties, since large changes in the residual
stress, adhesion, porosity, and hardness of the plate can be produced. ^
Surface Residual Stress
The formation of a favorable compressive residualstress pattern at the
surface is probably the most effective method of increasing fatigue per
formance. The subject of residual stress will be considered in greater
detail in Chap. 15. However, for the present discussion, it can be con
sidered that residual stresses are lockedin stresses which are present in
a part which is not subjected to an external force. Only macrostresses,
which act over regions which are large compared with the grain size, are
considered here. They can be measured by Xray methods or by noting
the changes in dimensions when a thin layer of material is removed from
the surface. Residual stresses arise when plastic deformation is not uni
form throughout the entire cross section of the part being deformed.
Consider a metal specimen where the surface has been deformed in
tension by bending so that part of it has undergone plastic deformation.
When the external force is removed, the regions which have been plasti
cally deformed prevent the adjacent elastic regions from undergoing com
plete elastic recovery to the unstrained condition. Thus, the elastically
1 "Fatigue Durability of Carburized Steel," American Society for Metals, Metals
Park, Ohio, 1957.
2 A detailed review of the effect of electroplating on fatigue strength is given by
R. A. R. Hammond and C. WiUiams, Met. Reviews, vol. 5, pp. 165223, 1960.
318 Applications to Materials Testing
[Chap. 12
deformed regions are left in residual tension, and the regions which were
plastically deformed must be in a state of residual compression to balance
the stresses over the cross section of the specimen. In general, for a
situation where part of the cross section is deformed plastically while the
rest undergoes elastic deformation, the region which was plastically
Tension
Connpression ^crmo
^^/?
^l
^o>
id)
Fig. 1 29. Superposition of applied and residual stresses.
deformed in tension will have a compressive residual stress after unload
ing, while the region which was deformed plastically in compression will
have a tensile residual stress when the external force is removed. The
maximum value of residual stress which can be produced is equal to the
elastic limit of the metal.
For many purposes residual stresses can be considered identical to the
stresses produced by an external force. Thus, the addition of a com
pressive residual stress, which exists at a point on the surface, to an
externally applied tensile stress on that surface decreases the likelihood
of fatigue failure at that point. Figure 129 illustrates this effect. Fig
ure 129a shows the elasticstress distribution in a beam with no residual
stress. A typical residualstress distribution, such as would be produced
Sec. 129] Fatigue of Metals 319
by shot peening, is shown in Fig. 1296. Note that the high compressive
residual stresses at the surface must be balanced by tensile residual
stresses over the interior of the cross section. In Fig. 129c the stress
distribution due to the algebraic summation of the external bending
stresses and the residual stresses is shown. Note that the maximum
tensile stress at the surface is reduced by an amount equal to the surface
compressive residual stress. The peak tensile stress is displaced to a
point in the interior of the specimen. The magnitude of this stress
depends on the gradient of applied stress and the residualstress distri
bution. Thus, subsurface initiation of failure is possible under these
conditions. It should also be apparent that the improvements in fatigue
performance which result from the introduction of surface compressive
residual stress will be greater when the loading is one in which a stress
gradient exists than when no stress gradient is present. However, some
improvement in the fatigue performance of axial loaded fatigue speci
mens results from surface compressive residual stresses, presumably
because the surface is such a potential source of weakness.
The chief commercial methods of introducing favorable compressive
residual stresses in the surface are by surface rolling with contoured
rollers and by shot peening. Although some changes in the strength of
the metal due to strain hardening occur during these processes, it is
believed that the improvement in fatigue performance is due chiefly to
the formation of surface compressive residual stress. Surface rolling is
particularly adapted to large parts. It is frequently used in critical
regions such as the fillets of crankshafts and the bearing surface of rail
road axles. Shot peening consists in projecting fine steel or castiron shot
against the surface at high velocity. It is particularly adapted to mass
produced parts of fairly small size. The severity of the stress produced
by shot peening is frequently controlled by measuring the residual defor
mation of shotpeened beams called Almen strips. The principal varia
bles in this process are the shot velocity and the size, shape, and hardness
of the shot. Care must be taken to ensure uniform coverage over the
area to be treated. Frequently an additional improvement in fatigue
properties can be obtained by carefully polishing the shotpeened surface
to reduce the surface roughness. Other methods of introducing surface
compressive residual stresses are by means of thermal stresses produced
by quenching steel from the tempering temperature and from stresses
arising from the volume changes accompanying the metallurgical changes
resulting from carburizing, nitriding, and induction hardening.
It is important to recognize that improvements in fatigue properties
do not automatically result from the use of shot peening or surface roll
ing. It is possible to damage the surface by excessive peening or rolling.
Experience and testing are required to establish the proper conditions
320 Applications to Materials Testing [Chap. 12
which produce the optimum residualstress distribution. Further, cer
tain metallurgical processes yield surface tensile residual stresses. Thus,
surface tensile stresses are produced by quenching deephardening steel,
and this stress pattern may persist at low tempering temperatures.
Grinding of hardened steel requires particular care to prevent a large
decrease in fatigue properties. It has been shown ^ that either tensile or
compressive surface residual stresses can be produced, depending upon
the grinding conditions. Further, the polishing^ methods ordinarily used
for preparing fatigue specimens can result in appreciable surface residual
stress. It is quite likely that lack of control of this factor in specimen
preparation is responsible for much of the scatter in fatiguetest results.
It is important to realize that residualstress patterns may be modified
by plastic deformation or by thermal activation. Thus, it is possible for
periods of overload or periods of increased temperature to result in some
relief of residual stress. The data on "fading" of residual stress are very
meager and not too reliable. In general, while the possibility of fading
of residual stress during service should be recognized, it does not prohibit
the use of compressive residual stress as the most effective method of
combating fatigue failure.
1210. Corrosion Fatisue
The simultaneous action of cyclic stress and chemical attack is known
as corrosion fatigue.^ Corrosive attack without superimposed stress
often produces pitting of metal surfaces. The pits act as notches and
produce a reduction in fatigue strength. However, when corrosive attack
occurs simultaneously with fatigue loading, a very pronounced reduction
in fatigue properties results which is greater than that produced by prior
corrosion of the surface. When corrosion and fatigue occur simultane
ously, the chemical attack greatly accelerates the rate at which fatigue
cracks propagate. Materials which show a definite fatigue limit when
tested in air at room temperature show no indication of a fatigue limit
when the test is carried out in a corrosive environment. While ordinary
fatigue tests in air are not affected by the speed of testing, over a range
from about 1,000 to 12,000 cycles/min, when tests are made in a corrosive
environment there is a definite dependence on testing speed. Since cor
rosive attack is a timedependent phenomenon, the higher the testing
speed, the smaller the damage due to corrosion. Corrosionfatigue tests
may be carried out in two ways. In the usual method the specimen is
iL. P. Tarasov, W. S. Hyler, and H. R. Letner, Proc. ASTM, vol. 57, pp. 601622,
1957.
2 An extensive review of the literature on this subject has been prepared by P. T.
Gilbert, Met. Reviews, vol. 1, pp. 379417, 1956.
)CC.
1210] Fatigue of Metals 321
continuously subjected to the combined influences of corrosion and cyclic
stress until failure occurs. In the twostage test the corrosion fatigue
test is interrupted after a certain period and the damage which was pro
duced is evaluated by determining the remaining life in air. Tests of the
last type have helped to establish the mechanism of corrosion fatigue.^
The action of the cyclic stress causes localized disruption of the surface
oxide film so that corrosion pits can be produced. Many more small pits
occur in corrosion fatigue than in corrosive attack in the absence of stress.
The cyclic stress will also tend to remove or dislodge any corrosion prod
ucts which might otherwise stifle the corrosion. The bottoms of the pits
are more anodic than the rest of the metal so that corrosion proceeds
inward, aided by the disruption of the oxide film by cyclic strain. Crack
ing will occur when the pit becomes sharp enough to produce a high stress
concentration.
There is evidence to indicate that even fatigue tests in air at room
temperature are influenced by corrosion fatigue. Fatigue tests on copper
showed that the fatigue strength was higher in a partial vacuum than in
air."^ Separate tests in oxygen and water vapor showed little decrease
over the fatigue strength in vacuum. It was concluded that water vapor
acts as a catalyst to reduce the fatigue strength in air, indicating that
the relative humidity may be a variable to consider in fatigue testing.
Subsequent work with copper^ showed that the fatigue life was much
longer in oxygenfree nitrogen than in air. Metallographic observation
showed that the development of persistent slip bands was slowed down
when tests were made in nitrogen.
A number of methods are available for minimizing corrosionfatigue
damage. In general, the choice of a material for this type of service
should be based on its corrosionresistant properties rather than the con
ventional fatigue properties. Thus, stainless steel, bronze, or beryllium
copper would probably give better service than heattreated steel. Pro
tection of the metal from contact with the corrosive environment by
metallic or nonmetallic coatings is successful provided that the coating
does not become ruptured from the cyclic strain. Zinc and cadmium
coatings on steel and aluminum coatings on Alclad aluminum alloys are
successful for many corrosionfatigue applications, even though these
coatings may cause a reduction in fatigue strength when tests are con
ducted in air. The formation of surface compressive residual stresses
tends to keep surface notches from opening up and giving ready access
to the corrosive medium. Nitriding is particularly effective in combating
corrosion fatigue, and shot peening has been used with success under cer
1 U. R. Evans and M. T. Simnad, Proc. Roy. Soc. (London), vol. 188A, p. 372, 1947.
2 H. J. Gough and D. G. Sopwith, J. Inst. Metals, vol. 72, pp. 415421, 1946.
3 N. Thompson, N. Wadsworth, and N. Louat, Phil. Mag., vol. 1, pp. 113126, 1956.
322 Applications to Materials Testing [Chap. 1 2
tain conditions. In closed systems it is possible to reduce the corrosive
attack by the addition of a corrosion inhibitor. Finally, the elimination
of stress concentrators by careful design is very important when corrosion
fatigue must be considered.
Fretting
Fretting is the surface damage which results when two surfaces in con
tact experience slight periodic relative motion. The phenomenon is more
related to wear than to corrosion fatigue. However, it differs from wear
by the facts that the relative velocity of the two surfaces is much lower
than is usually encountered in wear and that since the two surfaces are
never brought out of contact there is no chance for the corrosion products
to be removed. Fretting is frequently found on the surface of a shaft
with a pressfitted hub or bearing. Surface pitting and deterioration
occur, usually accompanied by an oxide debris (reddish for steel and
black for aluminum). Fatigue cracks often start in the damaged area,
although they may be obscured from observation by the surface debris.
Fretting is caused by a combination of mechanical and chemical effects.
Metal is removed from the surface either by a grinding action or by the
alternate welding and tearing away of the high spots. The removed
particles become oxidized and form an abrasive powder which continues
the destructive process. Oxidation of the metal surface occurs and the
oxide film is destroyed by the relative motion of the surfaces. Although
oxidation is not essential to fretting, as is demonstrated by relative motion
between two nonoxidizing gold surfaces, when conditions are such that
oxidation can occur fretting damage is many times more severe.
There are no completely satisfactory methods of preventing fretting.
If all relative motion is prevented, then fretting will not occur. Increas
ing the force normal to the surfaces may accomplish this, but the damage
increases with the normal force up to the point where relative motion is
stopped. If relative motion cannot be completely eliminated, then reduc
tion of the coefficient of friction between the mating parts may be bene
ficial. Solid lubricants such as MoS are most successful, since the chief
problem is maintaining a lubricating film for a long period of time.
Increasing the wear resistance of the surfaces so as to reduce surface
welding is another approach. Exclusion of the atmosphere from the
two surfaces will reduce fretting, but this is frequently difficult to do
with a high degree of effectiveness. Several excellent reviews of this
subject have been published.^'
1 R. B. Waterhouse, Proc. Inst. Mech. Encjrs. {London), vol. 169, pp. 11571172,
1955.
2 P. L. Teed, Met. Reviews, vol. 5, pp. 267295, 1960.
1211
Fatigue of Metals
323
^;^,^
fO.J
\\\^ /? =
V^ >? =
0.3
1 '
1.0
!
10'
10^
Cycles to failure
(a)
10'
1211. Effect of Mean Stress on Fatigue
Most of the fatigue data in the Uterature have been determined for
conditions of completely reversed cycles of stress, o^ = 0. However,
conditions are frequently met in engineering practice where the stress
situation consists of an alternating
stress and a superimposed mean, or
steady, stress. The possibility of
this stress situation has already been
considered in Sec. 122, where var
ious relationships between a^ and
a a have been given.
There are several possible methods
of determining an *SA^ diagram for
a situation where the mean stress
is not equal to zero. Figure 1210
shows the two most common meth
ods of presenting the data. In Fig.
12lOa the maximum stress is plot
ted against log A'' for constant values
of the stress ratio R = a^i^/cnm^.
This is achieved by applying a series
of stress cycles with decreasing maxi
mum stress and adjusting the mini
mum stress in each case so that it is
a constant fraction of the maxi
mum stress. The case of com
pletely reversed stress is given at
R = 1.0. Note that as R be
comes more positive, which is
equivalent to increasing the mean
stress, the measured fatigue limit
becomes greater. Figure 12105
shows the same data plotted in
terms of the alternating stress vs.
cycles to failure at constant values
of mean stress. Note that as the
mean stress becomes more positive the allowable alternating stress
decreases. Other ways of plotting these data are maximum stress vs.
cycles to failure at constant mean stress and maximum stress vs. cycles
to failure at constant minimum stress.
For each value of mean stress there is a different value of the limiting
range of stress, (7„ax — cr,„,n which can be withstood without failure.
Fig. 1210. Two methods of plotting
fatigue data when the mean stress is
not zero.
324 Applications to Materials Testing
[Chap. 12
Early contributions to this problem were made by Goodman,^ so that
curves which show the dependence of limiting range of stress on mean
stress are frequently called Goodman diagrams. Figure 1211 shows one
common type of Goodman diagram which can be constructed from fatigue
data of the type illustrated in Fig. 1210. Basically, this diagram shows
the variation of the limiting range of stress, omax — o'min with mean stress.
Note that as the mean stress becomes more tensile the allowable range of
stress is reduced, until at the tensile strength d„ the stress range is zero.
Compression^
> Tension
Mean stress ct^^j ^
Completely reversed stress
data plotted here
Fig. 1 21 1 . Goodman diagram.
However, for practical purposes testing is usually stopped when the yield
stress (To is exceeded. The test data usually lie somewhat above and
below the omax and a^^^ lines, respectively, so that these lines shown on
Fig. 1211 may actually be curves. A conservative approximation of the
Goodman diagram may be obtained, in lieu of actual test data, by draw
ing straight lines from the fatigue limit for completely reversed stress
(which is usually available from the literature) to the tensile strength.
A diagram similar to Fig. 1211 may be constructed for the fatigue
strength at any given number of cycles. Very few test data exist for
conditions where the mean stress is compressive. Data^ for SAE 4340
steel tested in axial fatigue indicate that the allowable stress range
increases with increasing compressive mean stress up to the yield stress
^ John Goodman, "Mechanics Applied to Engineering," 9th ed., Longmans, Green
& Co., Inc., New York, 1930.
2 J. T. Ransom, discussion in Proc. ASTM, vol. 54, pp. 847848. 1954.
Sec. 1211]
Fatigue of Metals
325
in compression. This is in agreement with the fact that compressive
residual stress increases the fatigue limit.
An alternative method of presenting meanstress data is shown in Fig.
1212. This is sometimes known as the HaigSoderberg diagram. ^ The
alternating stress is plotted against the mean stress. A straightline
relationship follows the suggestion of Goodman, while the parabolic curve
was proposed by Gerber. Test data for ductile metals generally fall
closer to the parabolic curve. However, because of the scatter in the
results and the fact that tests on notched specimens fall closer to the
Gerber parabola
Goodman line
Compression
Fig. 1212. Alternative method of plotting the Goodman diagram.
Goodman line, the linear relationship is usually preferred in engineering
design. These relationships may be expressed by the following equation,
Co — Ce
fe)'
(1210)
where a: = 1 for the Goodman line, .r = 2 for the Gerber parabola, and
Oe is the fatigue limit for completely reversed loading. If the design is
based on the yield strength, as indicated by the dashed Soderberg line in
Fig. 1212, then ao should be substituted for o„ in Eq. (1210).
Figure 1212 is obtained for alternating axial or bending stresses with
static tension or compression or alternating torsion with static tension.
However, for alternating torsion with static torsion or alternating bend
ing with static torsion there is no effect of the static stress on the allow
able range of alternating stress provided that the static yield strength is
not exceeded. Sines has shown that these results can be rationalized
1 C. R. Soderberg, Trans. ASME, vol. 52, APM522, 1930.
2 G. Sines, Failure of Materials under Combined Repeated Stresses with Super
imposed Static Stresses, NACA Tech. Note 3495, 1955.
326 Applications to Materials Testing [Chap. 12
in the following manner : The planes of maximum alternating shear stress
are determined, and the static normal stresses Si and S2 on these planes
are established. When *Si + S2 is positive, an increase in static stress
reduces the permissible value of aa. When Si + S2 is negative, the per
missible value of aa is increased. Finally, when Si + ^2 is zero, the static
stress has no effect on Ca regardless of the applied static stress.
1 21 2. Fatisue under Combined Stresses
Many machine parts must withstand complex loadings with both
alternating and steady components of stress. Ideally, it should be possi
ble to predict the fatigue performance of a combined stress system by
substituting the fatigue strength for simple types of loading into the
equation of the failure criterion, just as the yield stress in tension can be
used with the distortionenergy criterion of failure to predict static yield
ing under a complex state of stress. Although the data on combined
stress fatigue failure are fewer and less reliable than for static yielding,
certain generalizations can be made. Fatigue tests with different
combinations of bending and torsion ^ show that for ductile metals the
distortionenergy criterion provides the best overall fit. For brittle
materials a maximum principal stress theory provides the best criterion
of failure.
Sines^ has suggested a failure criterion that accounts for the effect of
combined stresses and the influence of a static mean stress.
4= [(^1  ^2)2 + (cr2  a^y + (c73  ai)Y' = <^e  C^iS, + Sy + S,)
V2
(1211)
where ai, a 2, 03 = alternating principal stresses
Sx, Sy, Sz = static stresses
(Te = fatigue strength for completely reversed stress
C2 = a material constant giving variation of aa versus am',
i.e., it is the slope of the Goodman line in Fig. 1212
Fatigue failure will occur if the left side of Eq. (1211) is greater than the
right side. The constants ae and C2 can be evaluated at any value of
cycles to failure. The effect of residual stresses can be introduced by
adding these terms to the static stresses on the right side of the equation.
Thus, Eq. (1211) shows that compressive residual stress will allow a
greater alternating stress for the same fatigue life.
1 H. J. Gough, Proc. Inst. Mech. Engrs. (London), vol. 160, pp. 417440, 1949;
W. N. Findley and P. N. Mathur, Proc. Soc. Exptl. Stress Anal, vol. 14, no. 1, pp. 35
46, 1956.
" Sines, op. cit.
Sec. 1213]
Fatigue of Metals 327
Prestress
1213. Overstressing and Understressing
The conventional fatigue test subjects a specimen to a fixed stress
amplitude until the specimen fails. Tests may be made at a number of
different values of stress to determine the SN curve, but each time the
maximum stress is held constant until the test is completed. However,
there are many practical applications where the cyclic stress does not
remain constant, but instead there are periods when the stress is either
above or below some average design stress level. Further, in other appli
cations complex loading conditions,
such as are illustrated in Fig. 122c,
are encountered. For these condi
tions it is difficult to arrive at an
average stress level, and the load
ing cannot be assumed to vary
sinusoidally. Special fatigue tests
which apply a random load have
been developed for these cases. ^
Overstressing is the process of
testing a virgin specimen for some
number of cycles less than that re
quired for failure, at a stress above
the fatigue limit, and then subse
quently running the specimen to
failure at another stress. The initial stress to which the specimen is sub
jected is called the prestress, while the final stress is the test stress. The
ratio of the number of cycles of overstress at the prestress to the mean vir
gin fatigue life at this same stress is called the cycle ratio. The damage
produced by a given cycle ratio of overstress is frequently evaluated by the
reduction in fatigue life at the test stress. Thus, referring to Fig. 1213,
for a cycle ratio at the prestress of AB/ AC the damage at the test stress
is EF/DF. EF represents the amount which the virgin fatigue life at
the test stress has been reduced by the overstress. Extensive experience^
with this type of test has shown that a high prestress followed by a lower
test stress causes a higher percentage of damage at the test stress than
the reduction in life used up by the cycle ratio. On the other hand, for a
given cycle ratio at a prestress lower than the test stress, the reduction in
life at the test stress will usually be a lower percentage than the cycle
ratio. The higher the value of the test stress, for a constant difference
between prestress and test stress, the more nearly the damage at the test
stress equals the cycle ratio at the prestress.
1 A. K. Head, Proc. Inter. Conf. Fatigue of Metals, London, 1956, pp. 301303.
2 J. B. Kommers. Proc. ASTM, vol. 45, pp. 532541, 1945.
log /!/
Fig. 1 21 3. Test procedure for determin
ing fatigue damage produced by over
stressing.
328 Applications to Materials Testing [Chap. 1 2
Another way of measuring the damage due to overstressing is to sub
ject a number of specimens to a certain cycle ratio at the prestress and
then determine the fatigue Hmit of the damaged specimens. Bennett^
has shown that increasing the cycle ratio at the prestress produces a
greater decrease in the fatigue limit of damaged specimens, while similar
experiments^ which employed a statistical determination of the fatigue
limit showed a much greater reduction in the fatigue limit due to the
overstressing. In line with this point, it is important to note that,
because of the statistical nature of fatigue, it is very difficult to reach
reliable conclusions by overstressing tests, unless statistical methods are
used.
If a specimen is tested below the fatigue limit, so that it remains
unbroken after a long number of cycles, and if then it is tested at a
higher stress, the specimen is said to have been under stressed. Under
stressing frequently results in either an increase in the fatigue limit or
an increase in the number of cycles of stress to fracture over what would
be expected for virgin specimens. It has frequently been considered
that the improvements in fatigue properties due to understressing are
due to localized strain hardening at sites of possible crack initiation. A
different interpretation of the understressing effect has resulted from
experiments on the statistical determination of the fatigue limit. ^ Speci
mens which did not fail during the determination of the fatigue limit
showed greater than normal fatigue lives when retested at a higher stress.
By means of statistical analysis it was possible to show that the observed
lives at the higher stress were to be expected owing to the elimination of
the weaker specimens during the prior testing below the fatigue limit.
Thus, it was concluded that understressing was at least partly due to a
statistical selectivity effect.
If a specimen is tested without failure for a large number of cycles
below the fatigue limit and the stress is increased in small increments
after allowing a large number of cycles to occur at each stress level, it is
found that the resulting fatigue limit may be as much as 50 per cent
greater than the initial fatigue limit. This procedure is known as coaxing.
An extensive investigation of coaxing"* showed a direct correlation between
a strong coaxing effect and the ability for the material to undergo strain
aging. Thus, mild steel and ingot iron show a strong coaxing effect, while
brass, aluminum alloys, and heattreated lowalloy steels show little
improvement in properties from coaxing.
1 J. A. Bennett, Proc. ASTM, vol. 46, pp. 693714, 1946.
2 G. E. Dieter, G. T. Home, and R. F. Mehl, NACA Tech. Note 3211, 1954.
3 E. Epremian and R. F. Mehl, ASTM Spec. Tech. Publ. 137, 1952.
' G. M. Sinclair, Proc. ASTM, vol. 52, pp. 743758, 1952.
Sec. 1214] Fatigue of Metals 329
1214. EFfect of Metallursical Variables on Fatigue Properties
The fatigue properties of metals appear to be quite structuresensitive.
However, at the present time there are only a limited number of ways in
which the fatigue properties can be improved by metallurgical means.
By far the greatest improvements in fatigue performance result from
design changes which reduce stress concentrations and from the intelli
gent use of beneficial compressive residual stress, rather than from a
change in material. Nevertheless, there are certain metallurgical fac
tors which must be considered to ensure the best fatigue performance
from a particular metal or alloy. Fatigue tests designed to measure the
effect of some metallurgical variable, such as special heat treatments, on
fatigue performance are usually made with smooth, polished specimens
under completely reversed stress conditions. It is usually assumed that
any changes in fatigue properties due to metallurgical factors will also
occur to about the same extent under more complex fatigue conditions,
as with notched specimens under combined stresses. That this is not
always the case is shown by the notchsensitivity results discussed
previously.
Fatigue properties are frequently correlated with tensile properties.
In general, the fatigue limit of cast and wrought steels is approximately
50 per cent of the ultimate tensile strength. The ratio of the fatigue
limit (or the fatigue strength at 10^ cycles) to the tensile strength is called
the fatigue ratio. Several nonferrous metals such as nickel, copper, and
magnesium have a fatigue ratio of about 0.35. While the use of corre
lations of this type is convenient, it should be clearly understood that
these constant factors between fatigue limit and tensile strength are only
approximations and hold only for the restricted condition of smooth,
polished specimens which have been tested under zero mean stress at
room temperature. For notched fatigue specimens the fatigue ratio for
steel will be around 0.20 to 0.30.
Several parallels can be drawn between the effect of certain metallurgi
cal variables on fatigue properties and the effect of these same variables
on tensile properties. The effect of solidsolution alloying additions on
the fatigue properties of iron^ and aluminum^ parallels nearly exactly
their effect on the tensile properties. The fatigue strength of nonferrous
metals^ and annealed steel increases with decreasing grain size. Grain
size does not have an important influence on the unnotched fatigue
1 E. Epremian and E. F. Nippes, Trans. ASM, vol. 40, pp. 870896, 1948.
2 J. W. Riches, O. D. Sherby, and J. E. Dorn, Trans. ASM, vol. 44, pp. 882895,
1952.
3 G. M. Sinclair and W. J. Craig, Trans. ASM, vol. 44, pp. 929948, 1952.
330 Applications to Materials Testing
[Chap. 12
properties of heattreated steels. Gensamer^ showed that the fatigue
hmit of a eutectoid steel increased with decreasing isothermalreaction
temperature in the same fashion as did the yield strength and the tensile
strength. However, the greater structure sensitivity of fatigue proper
ties, compared with tensile properties, is shown in tests comparing the
fatigue limit of a plaincarbon eutectoid steel heattreated to coarse
pearlite and to spheroidite of the same tensile strength. ^ Even though
the steel in the two structural conditions had the same tensile strength,
the pearlitic structure resulted in a significantly lower fatigue limit due to
the higher notch effects of the carbide lamellae in pearlite.
In general, quenched and tempered microstructures result in the opti
mum fatigue properties in heattreated lowalloy steels. However, at a
hardness level above about Re 40 a bainitic structure produced by
austempering results in better fatigue properties than a quenched and
tempered structure with the same hardness.^ Electron micrographs indi
cate that the poor performance of the quenched and tempered structure
150
140
130
120
'no
100
90
80
70
60
50
40
/
' *^^^.
^•^ '^^^
^•v."
^
^
/
m
\
^\
1
\
J
^^
D
y
i
•  SAE 4063
A  SAE 5150
A  SAE 4052
o  SAE 4140
»  SAE 4340
/
e  5^
i
2340
20
30
40 50
Rockwell C hardness
60
Fig. 1214. Fatigue limit of alloy steels as a function of Rockwell hardness. (M. F.
Garwood, H. H. Zurburg, and M. A. Erickson, in ^^Interpretation of Tests and Correla
tion with Service," p. 12, American Society for Metals Metals Park, Ohio, 1951.)
1 M. Gensamer, E. B. Pearsall, W. S. Pellini, and J. R. Low, Jr., Trans. ASM,
vol. 30, pp. 9831020, 1942.
2 G. E. Dieter, R. F. Mehl, and G. T. Home, Trans. ASM, vol. 47, pp. 423439,
1955.
3 F. Borik and R. D. Chapman, Trans. ASM, vol. 53, 1961.
Sec. 1214]
Fatigue of Metah
331
i
a
nnl ^^
^\n*^
1
^v. ^ "~~~"^
RO
^*~^ X
• ~" — "
70
60
1 —
X
^ X
^— 1 • \
o S/IE 1340
•  SAE 4042
A  SAE 4340
50
D SAE 5140
X  AISITS SOB 40
1 ■
40
100
90
80 70 60 50 40
Percent martensite as quenched
30
20
Fig. 1215. Fatigue limit of heattreated steel as a function of per cent martensite in
asquenched steel. {F. Borik, R. D. Chapman, and W. E. Jominy, Trans. ASM,
vol. 50, p. 250, 1958.)
is the result of the stressconcentration effects of the thin carbide films
that are formed during the tempering of martensite. For quenched and
tempered steels the fatigue limit increases with decreasing tempering
temperature up to a hardness of Re 45 to Re 55, depending on the steel.'
Figure 1214 shows the results obtained for completely reversed stress
tests on smooth specimens. The fatigue properties at high hardness levels
are extremely sensitive to surface preparation, residual stresses, and
inclusions. The presence of only a trace of decarburization on the sur
face may drastically reduce the fatigue properties. As is shown by Fig.
1215, only a small amount of nonmartensitic transformation products
can cause an appreciable reduction in the fatigue limit.' The effects of
slack quenching would be expected to be greater at higher hardness levels.
The influence of small amounts of retained austenite on the fatigue
properties of quenched and tempered steels has not been well established.
The results, which are summarized in Fig. 1214, indicate that below a
tensile strength of about 200,000 psi the fatigue limits of quenched and
tempered lowalloy steels of different chemical composition are about
equivalent when the steels are tempered to the same tensile strength.
This generalization holds for steels made by the openhearth or electric
furnace processes and for fatigue properties determined in the longitudinal
^ M. F. Garwood, H. H. Zurburg, and M. A. Erickson, "Interpretation of Tests
and Correlation with Service," American Society for Metals, Metals Park, Ohio, 1951.
2 F. Borik, R. D. Chapman, and W. E. Jominy, Trans. ASM, vol. 50, pp. 242257,
1958.
332 Applications to Materials Testing
[Chap. 12
direction of wrought products. However, tests have shown^ that the
fatigue Hmit in the transverse direction of steel forgings may be only
60 to 70 per cent of the longitudinal fatigue limit. It has been estab
lished that practically all the fatigue failures in transverse specimens
start at nonmetallic inclusions. Nearly complete elimination of inclu
sions by vacuum melting produces a considerable increase in the trans
verse fatigue limit (Table 123). The low transverse fatigue limit in
Table 123
Influence of Inclusions on Fatigue Limit of SAE 4340 Steelj
Longitudinal fatigue limit, psi .
Transverse fatigue limit, psi . .
Ratio transverse/longitudinal.
Hardness, Re
Electricfurnace
melted
116,000
79,000
0.68
27
Vacuum
melted
139,000
120,000
0.86
29
t Determined in repeatedbending fatigue test {R = 0).
Trans. ASM, vol. 46, pp. 12541269, 1954.
Data from J. T. Ransom,
steels containing inclusions is generally attributed to stress concentration
at the inclusions, which can be quite high when an elongated inclusion
stringer is oriented transverse to the principal tensile stress. However,
the fact that nearly complete elimination of inclusions by vacuum melt
ing still results in appreciable anisotropy of the fatigue limit indicates
that other factors may be important. Further investigations^ of this
subject have shown that appreciable changes in the transverse fatigue
limit which cannot be correlated with changes in the type, number, or
size of inclusions are produced by different deoxidation practices. Trans
verse fatigue properties appear to be one of the most structuresensitive
^engineering properties.
1215. Effect of Temperature on Fatigue
Lowtemperature Fatigue
Fatigue tests on metals at temperatures below room temperature
^how that the fatigue strength increases with decreasing temperature.
Although steels become more notchsensitive in fatigue at low temper
atures, there is no evidence to indicate any sudden change in fatigue
properties at temperatures below the ductiletobrittle transition temper
1 J. T. Ransom and R. F. Mehl, Proc. ASTM, vol. 52, pp. 779790, 1952.
2 J. T. Ransom, Trans. ASM, vol. 46, pp. 12541269, 1954.
3 G. E. Dieter, D. L. Macleary, and J. T. Ransom, Factors Affecting Ductility and
Fatigue in Forgings, "Metallurgical Society Conferences," vol. 3, pp. 101142, Inter
science Publishers, Inc., New York, 1959.
Sec. 1215] Fatigue of Metals 333
atiire. The fact that fatigue strength exhibits a proportionately greater
increase than tensile strength with decreasing temperature has been inter
preted as an indication that fatigue failure at room temperature is associ
ated with vacancy formation and condensation.
Hightemperature Fatigue
In general, the fatigue strength of metals decreases with increasing
temperature above room temperature. An exception is mild steel, which
shows a maximum in fatigue strength at 400 to 600°F. The existence of
a maximum in the tensile strength in this temperature range due to strain
aging has been discussed previously. As the temperature is increased
well above room temperature, creep will become important and at high
temperatures (roughly at temperatures greater than half the melting
point) it will be the principal cause of failure. The transition from
fatigue failure to creep failure with increasing temperature will result in
a change in the type of failure from the usual transcrystalline fatigue
failure to the intercrystalline creep failure. At any given temperature
the amount of creep will increase with increasing mean stress.
Ferrous materials, which ordinarily exhibit a sharp fatigue limit in
roomtemperature tests, will no longer have a fatigue limit when tested
at temperatures above approximately 800°F. Also, fatigue tests at high
temperature will be dependent on the frequency of stress application.
It is customary to report the total time to failure as well as the number
of cycles to failure.
In general, the higher the creep strength of a material, the higher its
hightemperature fatigue strength. However, the metallurgical treat
ment which produces the best hightemperature fatigue properties does
not necessarily result in the best creep or stressrupture properties. This
has been shown by Toolin and MocheP in hightemperature tests of a
number of superalloys. Fine grain size results in better fatigue proper
ties at lower temperatures. As the test temperature is increased, the
difference in fatigue properties between coarse and fine grain material
decreases until at high temperatures, where creep predominates, coarse
grain material has higher strength. In general, wrought alloys show
somewhat superior fatigue resistance, while castings are often more
resistant to creep. Procedures which are successful in reducing fatigue
failures at room temperature may not be effective in hightemperature
fatigue. For example, compressive residual stresses may be annealed out
before the operating temperature is reached.
Thermal Fatigue
The stresses which produce fatigue failure at high temperature do not
necessarily need to come from mechanical sources. Fatigue failure can
1 P. R. Toolin and N. L. Mochel, Proc. ASTM, vol. 47, pp. 677694, 1947.
334 Applications to Materials Testing [Chap. 12
be produced by fluctuating thermal stresses under conditions where no
stresses are produced by mechanical causes. Thermal stresses result
when the change in dimensions of a member as the result of a temper
ature change is prevented by some kind of constraint. For the simple
case of a bar with fixed end supports, the thermal stress developed by a
temperature change AT is
a = aE AT (1212)
where a = linear thermal coefficient of expansion
E — elastic modulus
If failure occurs by one application of thermal stress, the condition is
called thermal shock. However, if failure occurs after repeated appli
cations of thermal stress, of a lower magnitude, it is called thermal fatigue.^
Conditions for thermalfatigue failure are frequently present in high
temperature equipment. Austenitic stainless steel is particularly sensi
tive to this phenomenon because of its low thermal conductivity and high
thermal expansion. Extensive studies of thermal fatigue in this material
have been reported.^ The tendency for thermalfatigue failure appears
related to the parameter a/k/Ea, where o/ is the fatigue strength at the
mean temperature and k is the thermal conductivity. A high value of
this parameter indicates good resistance to thermal fatigue. An excellent
review of the entire subject of hightemperature fatigue has been prepared
by Allen and Forrest.^
BIBLIOGRAPHY
Battelle Memorial Institute, "Prevention of Fatigue of Metals," John Wiley & Sons,
Inc., New York, 1941. (Contains very complete bibliography up to 1941.)
Cazaud, R. : "Fatigue of Metals," translated by A. J. Fenner, Chapman & Hall, Ltd.,
London, 1953.
"The Fatigue of Metals," Institution of Metallurgists, London, 1955.
Freudenthal, A. M. (ed.): "Fatigue in Aircraft Structures," Academic Press, Inc.,
New York, 1956.
Grover, H. J., S. A. Gordon, and L. R. Jackson: "Fatigue of Metals and Structures,"
Government Printing Office, Washington, D.C., 1954. (Contains very com
plete compilation of fatigue data.)
Pope, J. A. (ed.): "Metal Fatigue," Chapman & Hall, Ltd., London, 1959.
Proceedings of the International Conference on Fatigue of Metals, London, 1956.
Sines, G., and J. L. Waisman (eds.): "Metal Fatigue," McGrawHill Book Company,
Inc., New York, 1959.
Thompson, N., and N. J. Wadsworth: Metal Fatigue, Advances inPhys., vol. 7, no. 25,
pp. 72169, January, 1958.
^ Failure of metals like uranium which have highly anisotropic thermalexpansion
coefficients mnder repeated heating and cooling is also called thermal fatigue.
2L. F. Coffin, Jr., Trans. ASME, vol. 76, pp. 931950, 1954.
" N. P. Allen and P. G. Forrest, Proc. Intern. Conf. Fatigue of Metals. London, 1956,
pp. 327340.
Chapter 13
CREEP AND STRESS RUPTURE
131. The Hightemperature Materials Problem
In several previous chapters it has been mentioned that the strength
of metals decreases with increasing temperature. Since the mobility of
atoms increases rapidly with temperature, it can be appreciated that
diffusioncontrolled processes can have a very significant effect on high
temperature mechanical properties. High temperature will also result in
greater mobility of dislocations by the mechanism of climb. The equi
librium concentration of vacancies likewise increases with temperature.
New deformation mechanisms may come into play at elevated temper
atures. In some metals the slip system changes, or additional slip sys
tems are introduced with increasing temperature. Deformation at grain
boundaries becomes an added possibility in the hightemperature defor
mation of metals. Another important factor to consider is the effect of
prolonged exposure at elevated temperature on the metallurgical stability
of metals and alloys. For example, coldworked metals will recrystallize
and undergo grain coarsening, while agehardening alloys may overage
and lose strength as the secondphase particles coarsen. Another impor
tant consideration is the interaction of the metal with its environment at
high temperature. Catastrophic oxidation and intergranular penetration
of oxide must be avoided.
Thus, it should be apparent that the successful use of metals at
high temperatures involves a number of problems. Greatly accelerated
alloydevelopment programs have produced a number of materials with
improved hightemperature properties, but the everincreasing demands
of modern technology require materials with even better hightemper
ature strength and oxidation resistance. For a long time the principal
hightemperature applications were associated with steam power plants,
oil refineries, and chemical plants. The operating temperature in equip
ment such as boilers, steam turbines, and cracking units seldom exceeded
1000°F. With the introduction of the gasturbine engine, requirements
developed for materials to operate in critically stressed parts, like turbine
335
336 Applications to Materials Testing [Chap. 13
buckets, at temperatures around 1500°F. The design of more powerful
engines has pushed this limit to around 1800°F. Rocket engines and
ballisticmissile nose cones present much greater problems, which can be
met only by the most ingenious use of the available hightemperature
materials and the development of still better ones. There is no question
that the available materials of construction limit rapid advancement in
hightemperature technology.
An important characteristic of hightemperature strength is that it
must always be considered with respect to some time scale. The tensile
properties of most engineering metals at room temperature are independ
ent of time, for practical purposes. It makes little difference in the
results if the loading rate of a tension test is such that it requires 2 hr or
2 min to complete the test. Further, in roomtemperature tests the
anelastic behavior of the material is of little practical consequence.
However, at elevated temperature the strength becomes very dependent
on both strain rate and time of exposure. A number of metals under
these conditions behave in many respects like viscoelastic materials. A
metal subjected to a constant tensile load at an elevated temperature will
creep and undergo a timedependent increase in length.
The tests which are used to measure elevatedtemperature strength
must be selected on the basis of the time scale of the service which the
material must withstand. Thus, an elevatedtemperature tension test
can provide useful information about the hightemperature performance
of a shortlived item, such as a rocket engine or missile nose cone, but it
will give only the most meager information about the hightemperature
performance of a steam pipeline which is required to withstand 100,000 hr
of elevatedtemperature service. Therefore, special tests are required to
evaluate the performance of materials in different kinds of hightemper
ature service. The creej) test measures the dimensional changes which
occur from elevatedtemperature exposure, while the stressrupture test
measures the effect of temperature on the longtime loadbearing charac
teristics. Other tests may be used to measure special properties such
as thermalshock resistance and stress relaxation. These hightemper
ature tests will be discussed in this chapter from two points of view.
The engineering significance of the information obtained from the tests
will be discussed, and information which is leading to a better under
standing of the mechanism of hightemperature deformation will be
considered.
132. The Creep Curve
The progressive deformation of a material at constant stress is called
creep. The simplest type of creep deformation is viscous flow. A mate
Sec. 132]
Creep and Stress Rupture 337
rial is said to experience pure viscous flow if the rate of shear strain is
proportional to the applied shearing stress.
^/«
(131)
When the proportionality between these two factors can be expressed
by a simple constant, the material is said to show Newtonian viscosity.
T = V
dy
It
(132)
where ?? is the coefficient of viscosity. Most liquids obey Newton's law
of viscosity, but it is only partially followed by metals.
Primary creep
I H<
Secondary creep
n
Tertlory creep
y Fracture
Time t
Fig. 1 31 . Typical creep curve showing the three stages of creep. Curve A, constant
load test; curve B, constantstress test.
To determine the engineering creep curve of a metal, a constant load is
applied to a tensile specimen maintained at a constant temperature, and
the strain (extension) of the specimen is determined as a function of time.
Although the measurement of creep resistance is quite simple in principle,
in practice it requires considerable laboratory equipment. ^^ The elapsed
time of such tests may extend to several months, while some tests have
been run for more than 10 years. The general procedures for creep test
ing are covered in ASTM Specification E13958T.
Curve A in Fig. 131 illustrates the idealized shape of a creep curve.
The slope of this curve (de/dt or e) is referred to as the creep rate. Follow
^ Creep and Creep Rupture Tests, ASM INIetals Handbook, Supplement, August,
1955, Metal Progr., vol. 68, no. 2A, pp. 175184, Aug. 15, 1955.
U. A. Fellows, E. Cook, and H. S. Avery, Trans. AIME, vol. 150, p. 358, 1942.
338
Applications to Materials Testing
[Chap. 1 3
ing an initial rapid elongation of the specimen, eo, the creep rate decreases
with time, then reaches essentially a steady state in which the creep rate
changes little with time, and finally the creep rate increases rapidly with
time until fracture occurs. Thus, it is natural to discuss the creep curve
in terms of its three stages. It should be noted, however, that the degree
to which these three stages are readily distinguishable depends strongly
on the applied stress and temperature.
In making an engineering creep test, it is usual practice to maintain
the load constant throughout the test. > Thus, as the specimen elongates
and decreases in crosssectional area, the axial stress increases. The
initial stress which was applied to the specimen is usually the reported
value of stress. Methods of compensating for the change in dimensions
of the specimen so as to carry out the creep test under constantstress
Creep curve
Sudden strain
Transient creep
Viscous creep
+
+
Time Time Time Time
Fig. 1 32. Andrade's analysis of the competing processes which determine the creep
conditions have been developed. ^'^ When constantstress tests are made,
it is frequently found that no region of accelerated creep rate occurs
(region III, Fig. 131) and a creep curve similar to B in Fig. 131 is
obtained. Accelerated creep is found, however, in constantstress tests
when metallurgical changes occur in the metal. Curve B should be con
sidered representative of the basic creep curve for a metal.
Andrade's pioneer work on creep^ has had considerable influence on
the thinking on this subject. He considers that the constantstress creep
curve represents the superposition of two separate creep processes which
occur after the sudden strain which results from applying the load. The
first component is a transient creep, which has a decreasing creep rate
with time. Added to this is a constantrate creep process called viscous
creep. The superposition of these creep processes is illustrated in Fig.
132. Andrade found that the creep curve could be represented by the
1 E. N. da C. Andrade and B. Chalmers, Proc. Roij Soc. (London), vol. 138A, p. 348,
1932.
2 R. L. Fullman, R. P. Carreker, and J. C. Fisher, Traris. AIME, vol. 3 97, pp. 657
659, 1953.
^ E. N. da C. Andrade, Proc. Roy. Soc. (London), vol. 90A, pp. 329342, 1914;
"Creep and Recovery," pp. 176198, American Society for Metals, Metals Park, Ohio,
1957.
Sec. 1 32] Creep and Stress Rupture 339
following empirical equation,
L = Lo(l + 131''^) exp Kt (133)
where L = length of specimen at time t
Lo, /3, K = empirically determined constants
The constant Lo approximates the length of the specimen when the sudden
strain produced by the application of the load has ceased. The transient
creep is represented by the constant /3. Thus, when k = 0, Eq. (133)
yields a creep rate which vanishes at long times.
L = Io(l + 0t''^)
■ "^^ = yiU&t^'^ (134)
dt
When /3 = 0,
L
y = exp Kt
dL
dt
kLq exp Kt = kL
The exponent k therefore represents an extension, per unit length, which
proceeds at a constant rate. It represents the viscous component of
creep. Strictly speaking, k represents quasiviscous flow because the
rate of change of length is not proportional to stress as required by Eq.
(132). Sometimes transient creep is referred to as /3 flow, and viscous
(steadystate) creep is referred to as k flow in keeping with Andrade's
analysis of the creep curve. Andrade's equation has been verified for
conditions extending up to several hundred hours, which result in total
extensions in excess of 1 per cent. Modifications of these equations will
be considered in another section of this chapter.
The various stages of the creep curve shown in Fig. 131 require fur
ther explanation. It is generally considered in this country that the
creep curve has three stages. In British terminology the instantaneous
strain designated by eo in Fig. 131 is often called the first stage of creep,
so that with this nomenclature the creep curve is considered to have four
stages. The strain represented by eo occurs practically instantaneously
on the application of the load. Even though the applied stress is below
the yield stress, not all the instantaneous strain is elastic. Most of this
strain is instantly recoverable upon the release of the load (elastic), while
part is recoverable with time (anelastic), and the rest is nonrecoverable
(plastic). Although the instantaneous strain is not really creep, it is
important because it may constitute a considerable fraction of the allow
able total strain in machine parts. Sometimes the instantaneous strain
340 Applications to Materials Testing
[Chap. 13
is subtracted from the total strain in the creep specimen to give the strain
due only to creep. This type of creep curve starts at the origin of
coordinates.
The first stage of creep, known as primary creep, represents a region of
decreasing creep rate. Primary creep is a period of predominantly tran
sient creep in which the creep resistance of the material increases by
virtue of its own deformation. For low temperatures and stresses, as in
the creep of lead at room temperature, primary creep is the predominant
creep process. The second stage of creep, known also as secondary creep,
is a period of nearly constant creep rate which results from a balance
between the competing processes
of strain hardening and recovery.
For this reason, secondary creep is
usually referred to as steadystate
creep. The average value of the
creep rate during secondary creep
is called the minimum creep rate.
Thirdstage or tertiary creep mainly
occurs in constantload creep tests
at high stresses at high tempera
tures. The reasons for the acceler
ated creep rate which leads to rapid
failure are not well known. It is
unlikely that tertiary creep is due
solely to necking of the specimen,
since many materials fail in creep
at strains which are too small to produce necking. Tertiary creep is more
probably the result of structural changes occurring in the metal. Evi
dence has been found for void formation and extensive crack formation
during this stage.
Figure 133 shows the effect of applied stress on the creep curve at
constant temperature. It is apparent that a creep curve with three well
defined stages will be found for only certain combinations of stress and
temperature. A similar family of curves is obtained for creep at con
stant stress for different temperatures. The higher the temperature, the
greater the creep rate. The basic difference for this case would be that
all the curves would originate from the same point on the strain axis.
The minimum creep rate is the most important design parameter
derived from the creep curve. Two standards of this parameter are
commonly used in this country, (1) the stress to produce a creep rate of
0.0001 per cent/hr or 1 per cent/10,000 hr, or (2) the stress for a creep
rate of 0.00001 per cent/hr or 1 per cent/ 100,000 hr (about 11 ^ years).
The first criterion is more typical of the requirements for jetengine alloys,
while the last criterion is used for steam turbines and similar equipment.
Time, hr
Fig. 133. Schematic representation of
effect of stress on creep curves at con
stant temperature.
133]
Creep and Stress Rupture 341
A loglog plot of stress vs. minimum creep rate frequently results in a
straight line. This type of plot is very useful for design purposes, and
its use will be discussed more fully in a later part of this chapter.
133. The Stressrupture Test
The stressrupture test is basically similar to the creep test except that
the test is always carried out to the failure of the material. Higher loads
are used with the stressrupture test than in a creep test, and therefore
the creep rates are higher. Ordinarily the creep test is carried out at
relatively low stresses so as to avoid tertiary creep. Emphasis in the
creep test is on precision determination of strain, particularly as to the
determination of the minimum creep rate. Creep tests are frequently
conducted for periods of 2,000 hr and often to 10,000 hr. In the creep
test the total strain is often less than 0.5 per cent, while in the stress
rupture test the total strain may be around 50 per cent. Thus, simpler
strainmeasuring devices, such as dial gages, can be used. Stressrupture
equipment is simpler to build, maintain, and operate than creeptesting
equipment, and therefore it lends itself more readily to multiple testing
units. The higher stresses and
creep rates of the stressrupture test
cause structural changes to occur in
metals at shorter times than would
be observed ordinarily in the creep
test, and therefore stressrupture
tests can usually be terminated in
1,000 hr. These factors have con
tributed to the increased use of the
stressrupture test. It is partic
ularly well suited to determin
ing the relative hightemperature
strength of new alloys for jetengine
applications. Further, for appli
cations where creep deformation can be tolerated but fracture must be
prevented, it has direct application in design.
The basic information obtained from the stressrupture test is the time
to cause failure at a given nominal stress for a constant temperature.
The elongation and reduction of area at fracture are also determined.
If the test is of suitable duration, it is customary to make elongation
measurements as a function of time and from this to determine the mini
mum creep rate. The stress is plotted against the rupture time on a
loglog scale (Fig. 134). A straight line will usually be obtained for
each test temperature. Changes in the slope of the stressrupture line
are due to structural changes occurring in the material, e.g., changes from
0.001 001
,000
Rupture time, tir
Fig. 1 34. Method of plotting stress
rupture data (schematic).
342 Applications to Materials Testing [Chap. 1 3
transgranular to intergraniilar fracture, oxidation, recrystallization and
grain growth, or other structural changes such as spheroidization, graphi
tization, or sigmaphase formation. It is important to know about the
existence of such instabilities, since serious errors in extrapolation of the
data to longer times can result if they are not detected.
1 34. Deformation at Elevated Temperature
The principal deformation processes at elevated temperature are slip,
subgrain formation, and grainboundary sHding. Hightemperature
deformation is characterized by extreme inhomogeneity. Measure
ments of local creep elongation^ at various locations in a creep specimen
have shown that the local strain undergoes many periodic changes with
time that are not recorded in the changes in strain of the total gage
length of the specimen. In largegrained specimens, local regions may
undergo lattice rotations which produce areas of misorientation.
A number of secondary deformation processes have been observed in
metals at elevated temperature. These include multiple slip, the for
mation of extremely coarse slip bands, kink bands, fold formation at
grain boundaries, and grainboundary migration. Many of the defor
mation studies at elevated temperature have been made with largegrain
size sheet specimens of aluminum. (Aluminum is favored for this type
of study because its thin oxide skin eliminates problems from oxidation.)
Studies have also been made of creep deformation in iron, magnesium,
and lead. It is important to remember that all the studies of hightem
perature deformation have been made under conditions which give a
creep rate of several per cent in 100 or 1,000 hr, while for many engi
neering applications a creep rate of less than 1 per cent in 100,000 hr is
required. Because the deformation processes which occur at elevated
temperature depend on the rate of strain as well as the temperature, it is
not always possible to extrapolate the results obtained for high strain
rate conditions to conditions of greater practical interest. Much of the
work on deformation processes during creep has been reviewed by SuUy^
and Grant and Chaudhuri.^
Deformation hy Slip
New slip systems may become operative when metals are deformed at
elevated temperature. Slip occurs in aluminum* along the { lU } , { 100} ,
or {211} planes above 500°F. Zinc shps on the nonbasal {1010} planes
1 H. C. Chang and N. J. Grant, Trans. AIME, vol. 197, p. 1175, 1953.
2 A. H. Sully, "Progress in Metal Physics," vol. 6, pp. 135180, Pergamon Press,
Ltd., London, 1956.
3 N. J. Grant and A. R. Chaudhuri, Creep and Fracture, in "Creep and Recovery,"
pp. 284343, American Society for Metals, Metals Park, Ohio, 1957.
* I. S. Servi. J. T. Norton, and N. J. Grant, Trans. AIME, vol. 194, p. 965, 1952.
Sec. 1 34] Creep and Stress Rupture 343
in the (1210) directions above 570°F, and there is evidence of nonbasal
hightemperature slip in magnesium. ^ The slip bands produced at high
temperature do not generally resemble the straight slip lines which are
usually found after roomtemperature deformation. Although hightem
perature slip may start initially as fairly uniformly spaced slip bands, as
deformation proceeds there is a tendency for further shear to be restricted
to a few of the slip bands. The tendency for cross slip and the formation
of deformation bands increases with temperature. Fine slip lines, which
are difficult to resolve with the optical microscope, have been found
between the coarse slip bands in creep specimens of aluminum.^ These
represent the traces of slip planes on which only very small amounts of
shear have occurred. The significance of fine slip to creep deformation
will be discussed later.
In one of the first investigations of creepdeformation processes, Hanson
and Wheeler^ established that the slipband spacing increases with either
an increase in temperature or a decrease in stress. Subsequent work
on aluminum and its alloys^' ^ showed that the slipband spacing was
inversely proportional to the applied stress but independent of temper
ature. These observations may be interpreted in the following way:
If aluminum with a certain initial grain size is tested at a certain stress,
there will be a certain characteristic slipband spacing. If the grain size
is smaller than the slipband spacing, the slip bands will not be visible
in the specimen after deformation. Deformation of the grains will occur
by shear along the grain boundaries and by the breakup of the grains into
"cells," or subgrains.^ Deformation at high temperatures and/or low
strain rates are conditions for which it is difficult to detect slip lines but
for which there is abundant evidence of grainboundary deformation.
This condition has often been called "slipless flow."
Complex deformation processes occur in the vicinity of the grain
boundaries. While grain boundaries restrict deformation at high tem
perature to a lesser extent than at room temperature, they still exert a
restraining influence on deformation.
Suhgrain Formation
Creep deformation is quite inhomogeneous, so that there are many
opportunities for lattice bending to occur. Kink bands, deformation
bands, and local bending near grain boundaries are known to occur.
1 A. R. Chaudhuri, H. C. Chang, and N. J. Grant, Trans. AIME, vol. 203, p. 682,
1955.
2 D. McLean, J. Inst. Metals, vol. 81, p. 133, 19521953.
3 D. Hanson and M. A. Wheeler, J. Inst. Metals, vol. 55, p. 229, 1931.
^ I. S. Servi and N. J. Grant, Trans. .4 /il/E, vol. 191, p. 917, 1951.
6 G. D. Gemmell and N. J. Grant, Trans. AIME, vol. 209, pp. 417423, 1957.
« W. A. Wood, G. R. Wilms, and W. A. Rachinger, J. Inst. Metals, vol. 79, p. 159,
1951.
344 Applications to Materials Testing [Chap. 13
Polygoiiization can take place concurrently with lattice bending because
dislocation climb can occur readily at high temperature (see Sec. 610).
The formation of cells, or subgrains, as creep progresses has been observed
by means of X rays and metallographic techniques. The size of the sub
grains depends on the stress and the temperature. Large subgrains, or
cells, are produced by high temperature and a low stress or creep rate.
The decreasing creep rate found during primary creep is the result of the
formation of more and more subgrains as creep continues. The increased
number of lowangle boundaries provides barriers to dislocation move
ment and results in a decrease in creep strain.
Grainboundary Deformation
It has already been shown in Sec. 85 that the grainboundary relax
ation which is measured by internal friction at elevated temperature
indicates that the grain boundaries have a certain viscous behavior under
these conditions. Therefore, it is not surprising that the grain boundaries
behave in a manner to indicate considerable mobility when creep is pro
duced at high temperature. The main grainboundary processes which
are observed in hightemperature creep are grainboundary sliding, grain
boundary migration, and fold formation.
Grainboundary sliding is a shear process which occurs in the direction
of the grain boundary. It is promoted by increasing the temperature
and/or decreasing the strain rate. The question whether the sliding
occurs along the grain boundary ^ as a bulk movement of the two grains
or in a softened area in each grain adjacent to the grain boundary ^ has
not been answered. Grainboundary shearing occurs discontinuously in
time, and the amount of shear displacement is not uniform along the
grain boundary. Although the exact mechanism is not known, it is clear
that grainboundary sliding is not due to simple viscous sliding of one
grain past another because it is preceded by appreciable amounts of
plastic flow in adjacent crystals.
Grainboundary migration is a motion of the grain boundary in a
direction which is inclined to the grain boundary. It may be considered
to be stressinduced grain growth. Grainboundary migration is a creep
recovery process which is important because it allows the distorted mate
rial adjacent to the grain boundary to undergo further deformation. The
wavy grain boundaries which are frequently observed during hightem
perature creep are a result of inhomogeneous grainboundary deformation
and grainboundary migration.
For grainboundary deformation to occur without producing cracks
at the grain boundaries, it is necessary to achieve continuity of strain
' H. C. Chang and N. J. Grant, Trans. AIMS, vol. 206, p. 169, 1956.
2 F. N. Rhines, W. E. Bond, and M. A. Kissel, Trans. ASM, vol. 48, p. 919, 1956.
Sec. 1 35] Creep and Stress Rupture 345
along the grain boundary. A common method of accommodating grain
boundary strain at high temperature is by the formation of folds at the
end of a grain boundary. ^ Figure 135 shows a sketch of a fold.
The relative importance of slip and grainboundary displacement to
the total creep deformation has been investigated by McLean for alumi
num at 200°C. At this relatively low tem
perature he has shown that only a small
fraction of the total deformation is due to
grainboundary displacement, about half the
total deformation is due to slip which is
readily attributed to coarse slip bands, while
the remainder of the total deformation can
not be attributed to any microscopic defor
mation mechanism. McLean attributes this
"missing creep" to deformation by fine slip,
which is very difficult to detect with the I
microscope. It is believed that deformation
by fine slip can explain the observations Fig. 135. Fold formation at a
of earlier workers that creep deformation ^^P^^ P^^'^t (schematic).
occurs without slip (slipless flow). Greater
contribution to the total deformation from grainboundary displacement
would be expected at higher temperatures and lower stresses.
1 35. Fracture at Elevated Temperature
It has been known since the early work of Rosenhain and Ewen' that
metals undergo a transition from transgranular fracture to intergranular
fracture as the temperature is increased. When transgranular fracture
occurs, the slip planes are weaker than the grain boundaries, while for
intergranular fracture the grain boundary is the weaker component.
Jeffries'' introduced the concept of the equicohesive temperature (EOT),
which was defined as that temperature at which the grains and grain
boundaries have equal strength (Fig. 136a). Like the recrystallization
temperature, the ECT is not a fixed one. In addition to the effect of
stress and temperature on the ECT, the strain rate has an important
influence. Figure 1366 shows that decreasing the strain rate lowers the
ECT and therefore increases the tendency for intergranular fracture.
The effect of strain rate on the strengthtemperature relationship is
believed to be much larger for the grainboundary strength than for
1 H. C. Chang and N. J. Grant, Trans. AIME, vol. 194, p. 619, 1952.
2 D. McLean, J. Inst. Metals, vol. 80, p. 507, 19511952.
3 W. Rosenhain and D. Ewen, J. Inst. Metals, vol. 10, p. 119, 1913.
♦Z. Jeffries, Trans. AIME, vol. 60, pp. 474576, 1919.
346 Applications to Materials Testing
[Chap. 13
the strength of the grains. Since the amount of grainboundary area
decreases with increasing grain size, a material with a large grain size
will have higher strength above the
Grain boundary
ECT
Temperature
[a]
w ^Low strain rate
■<^^ y^High strain rate
t
^
^>^^r^~^^^v.
U'
\ — *
f*~~~— ^
n)
\
"x^ ^ ' ^^
CO
\
\
\
ECT, ECTg
Temperature — ^
Grain boundary
Grain
Range
of ECT
Pure metal
Temperature — ^
(f)
Fig. 1 36. The equicohesive temperature.
(After N. J. Grant, in " Utilization of Heat
Resistant Alloys," American Society for
Metals, Metals Park, Ohio, 1954.)
ECT than a fine grain material.
Below the ECT the reverse is true.
For metals and alloys of commercial
purity the ECT will occur within a
fairly sharp temperature interval.
However, for highpurity material
there is a wide range of temperature
where the strengths of the grains
and the grain boundaries are not
Fig. 1 37. Schematic drawings of the way
intergranular cracks form owing to grain
boundary sliding. (H. C. Chang and
N. J. Grant, Trans. AIME, vol. 206,
p. 545, 1956.)
very different^ (Fig. 136c) so that transgranular fracture can occur up
to rather high temperatures.
Under conditions where grain boundaries experience considerable
mobility, we have the situation of a thin, plastically weak region
embedded between two relatively strong matrix grains. This is a situ
ation that is conducive to grainboundary failure. While a number of
' Servi and Grant, op. cit., p. 909.
Sec. 1 36] Creep and Stress Rupture 347
reasonable models for grainboundary fracture have been suggested, none
are capable of predicting all the details of grainboundary fracture.
Two types of intergranular fracture have been observed under creep
conditions. Under conditions where grainboundary sliding can occur,
cracks may be initiated at triple points where three grain boundaries
meet on a plane of polish. This type of grainboundary failure is preva
lent for high stresses, where the total life is fairly short. Several methods
by which cracks form' as the result of grainboundary sliding are shown
schematically in Fig. 137. Zener has shown that large tensile stresses
should be developed at a triple point due to shear stresses acting along
the grain boundaries. When grainboundary migration and fold forma
tion can occur, the tendency for grainboundary fracture is diminished.
Grainboundary migration displaces the strained grain boundary to a
new unstrained region of the crystal, while the formation of folds per
mits the relief of stress concentration at grain corners by plastic defor
mation within the grains.
The second type of intergranular fracture is characterized by the
formation of voids at grain boundaries, particularly those which are
perpendicular to the tensile stress. The voids grow and coalesce into
grainboundary cracks. This type of fracture is most prevalent when
low stresses result in failure in relatively long times. At least two
mechanisms have been suggested for this type of fracture. One mecha
nism is based on the idea that the voids are formed by the condensation
of vacancies and grow by the diffusion of vacancies. Ballufh and Seigle*
have advanced a theory for the growth of voids based on the ideas used
to explain the sintering of metals. On the other hand, there are experi
ments" which show that grainboundary voids are not formed unless there
is grainboundary sliding. It is uncertain, at present, whether voids are
initiated at grain boundaries by a process of vacancy condensation or as
the result of localized plastic yielding.
1 36. Theories of Lowtemperature Creep
Creep is possible only because obstacles to deformation can be overcome
by the combined action of thermal fluctuations and stress. Diffusion
controlled processes are important chiefly at temperatures greater than
about onehalf the melting point. At lower temperatures recovery proc
1 H. C. Chang and N. J. Grant, Trans. AIME, vol. 206, pp. 544550, 1956.
' C. Zener, Micromechanism of Fracture, in "Fracturing of Metals," p. 3, American
Society for Metals, Metals Park, Ohio, 1948.
■" R. W. Balluffi and L. L. Seigle, Ada Met., vol. 5, p. 449, 1957.
'C. W. Chen and E. S. Machlin, Trans. AIME, vol. 209, pp. 829835, 1957; J.
Intrater and E. S. MachHn, Acta Met., vol. 7, p. 140, 1959.
348 Applications to Materials Testing [Chap. 1 3
esses which are not dependent on diffusion, such as cross slip, play impor
tant roles in the creep process. Hightemperature creep is predominantly
steadystate or viscous creep, while below Tml'^ transient, or primary,
creep predominates.
Andrade's equation for describing transient and steadystate creep was
discussed in Sec. 132. An alternative general equation for the time laws
of creep was suggested by Cottrell.^
e = At''' ' (136)
where A and n' are empirical constants. Different types of creep behav
ior are described by Eq. (136) depending upon the value of n' . If n' = 0,
the creep rate is constant and Eq. (136) represents steadystate creep.
When n' = 1, Eq. (136) becomes
e = a In f (137)
where a is a constant. This is the logarithmic creep law found at low
temperatures. 2 When n' = ^, Eq. (136) becomes Andrade's equation
for transient creep,
e = I3t'^^ (138)
Logarithmic creep occurs at low temperatures and low stresses, where
recovery cannot occur. It is believed to be a true exhaustion process in
which the ratedetermining step is the activation energy to move a dis
location. On the initial application of stress, the dislocations with the
lowest activation energy move first to produce the initial creep strain.
As these easytomove dislocations are exhausted, creep can continue only
by the movement of dislocations of higher activation energy. Therefore,
the activation energy for the process continuously increases, and the creep
rate decreases. Theoretical treatments of exhaustion creep that result in
a logarithmic equation have been proposed by Mott and Nabarro* and
Cottrell.4
Lowtemperature logarithmic creep obeys a mechanical equation of
state; i.e., the rate of strain at a given time depends only on the instan
taneous values of stress and strain and not on the previous strain history.
However, creep at higher temperatures is strongly dependent on prior
strain and thermal history and hence does not follow a mechanical equa
tion of state.
An exhaustion theory does not adequately describe the behavior during
1 A. H. Cottrell, /. Mech. and Phys. Solids, vol. 1, pp. 5363, 1952.
2 Logarithmic creep has been observed for copper below 200°K. [O. H. Wyatt, Proc.
Phys. Soc. (London), vol. 66B, p. 495, 1953].
5 N. F. Mott and F. R. N. Nabarro, "Report on Strength of Solids," p. 1, Physical
Society, London, 1948.
^ Cottrell, op. cit.
Sec. 1 37] Creep and Stress Rupture 349
transient creep. The decreasing creep rate during transient creep arises
from the increasing dislocation density and the formation of lowangle
boundaries. The recovery mechanisms operating during transient creep
are not well established. Analysis of existing data^ indicates that the
escape of screw dislocations from pileups by cross slip may be the chief
recovery mechanism in fee metals.
1 37. Theories of Hishtemperaturc Creep
Steadystate, or secondary, creep predominates at temperatures above
about T'm/2. Although there is some question whether a true steady
state condition is achieved for all combinations of stress and temperature,
there is ample experimental evidence to indicate that approximate steady
state conditions are achieved After a short period of testing in the high
temperature region. Steadystate creep arises because of a balance
between strain hardening and recovery. The effects of strain hardening
are recovered by the escape of screw dislocations from pileups by cross
slip and the escape of edge dislocations by climb. Since dislocation
climb has a higher activation energy than cross slip, it will be the rate
controlling step in steadystate creep.
Orowan first suggested that steadystate creep could be treated as a
balance between strain hardening and recovery. If is the slope of the
stressstrain curve at the applied stress a and r = da/dt is the rate of
recovery of the flow stress on annealing, then the steadystate condition
requires that the flow stress must remain constant.
d<r = ^dt\^de = (139)
ot oe
, = , = _fZ' = r (1310)
0(T/oe <p
where k is the steadystate creep rate (k flow) and r and ^ are defined
above. Cottrell and Aytekin^ have shown that the thermally activated
process of recovery can be expressed by
da —{H — qa) /iQ 1 1\
r= ^ = Cexp ^ (1311)
where H = an activation energy
q = an activation volume related to atomic stress concentration
C = a constant
k = Boltzmann's constant
1 G. Schoeck, Theory of Creep, in "Creep and Recovery," American Society of
Metals, Metals Park, Ohio, 1957.
2 E. Orowan, J. West Scot. Iron Steel Inst., vol. 54, pp. 4596, 1947.
3 A. H. Cottrell and V. Aytekin, J. Inst. Metals, vol. 77, p. 389, 1950.
350 Applications to Materials Testing [Chap. 1 3
Substituting Eq. (1311) into Eq. (1310) gives an equation for the steady
state creep rate.
. C (H  qa) .^_^,
^ = ^"^P kf (l^l^)
Theories of steadystate creep based on Eyring's chemical theory of
reaction rates were at one time very prominent in the thinking on this
subject. i'^ According to this concept, creep resulted from the shear of
"flow units" past each other through the periodic potential field of the
crystal lattice. The metal was assumed to behave like a viscous mate
rial, and for creep to occur the flow unit had to acquire sufficient energy
to surmount the potential barrier of the activated state. The theory
predicted a steadystate creep rate of the form
€ = C exp j^ sinh ^ (1313)
where AF is the free energy of activation and 5 is a constant describing
the size of the flow unit. The flow units were not defined in terms of
crystal structure, so that this theory had little physical basis. However,
it was an important advance in creep theory because it emphasized that
creep was a thermally activated process.
A relationship very similar to Eq. (1313) has been used by a number
of investigators for describing steadystate creep data of annealed metals
in terms of applied stress and temperature.^
e = ^oexp^ sinh^ (1314)
The parameters Ao, H, and q are independent of a, and only q varies sig
nificantly with temperature. If qa/kT is greater than 2, the above
equation simplifies to
e = HAo exp — ^^^^^ (1315)
Equation (1315) implies the existence of a linear relationship between
log e and (T. A linear relationship is also often obtained when log e is
plotted against logo. This is a conventional method of plotting engi
neering creep data. The corresponding equation
e = Ba' (1316)
is often used in the engineering analysis of the creep of structures and
machine elements.
1 W. Kauzmann, Trans. AIME, vol. 143, pp. 5783, 1941.
"^ S. Dushman, L. W. Dunbar, and H. Huthsteiner, /. Appl. Phys., vol. 15, p. 108,
1944.
3 P. Feltham and J. D. Meakin, Acta Mel., vol. 7, pp. 614627, 1959.
)ec.
137]
Creep and Stress Rupture 351
Dorn^'^ has made valuable contributions to the knowledge of high
temperature creep by putting the reactionrate theory of creep on a
better physical basis. In his creep experiments on highpurity poly
crystalline aluminum, careful attention has been given to changes in
lattice distortion and subgrain formation during the creep test. It has
" 0.60
0.50
0.40
0.30
0.20
0.10
t
Pure aluminum (99.987%)
Creep under constant stress of 3,000 psi
4x10"'^
e^te^"/RT, hr
Fig. 138. Creep strain vs. = t %x^ {—l^H /RT). {J. E. Dorn, in "Creep and
Recovery," American Society for Metals, Metals Park, Ohio, 1957.)
been found that creep can be correlated in terms of a temperature
compensated time parameter 6.
= t exp
AH,
RT
(1317)
where t = time of creep exposure
Ai/c = activation energy for creep
R = universal gas constant*
T = absolute temperature
For constant load or stress conditions, the total creep strain is a simple
function of d.
e = f(d,ac) o'c = constant (1318)
Figure 138 shows a typical correlation between e and d for creep data
obtained at several temperatures. An important characteristic of this
' J. E. Dorn, The Spectrum of Activation Energies for Creep, in "Creep and Recov
ery," pp. 255283, American Society for Metals, Metals Park, Ohio, 1957.
2 J. E. Dorn, /. Mech. and Phys. Solids, vol. 3, p. 85, 1954.
^ Boltzmann's constant A; and the gas constant R are related through the equation
R = kN, where A' is Avogadro's number.
352 Applications to Materials Testing [Chap. 13
correlation is that equivalent structure, as determined by metallography
and X rays, is obtained for the same values of e and 6.
If Eq. (1318) is differentiated with respect to time,
Evaluating Eq. (1319) at the minimum creep rate im gives
aTJ
im = ner.,a) exp ^ (1320)
However, dm is only a function of the creep stress ac, so that
<Tc = F (im exp H^ = F{Z) (1321)
Z is the ZenerHollomon parameter considered previously in Sec. 98.
Not only can this equation be used for describing the relation between
temperature and strain rate for the relatively high strain rates of the
tension test, but it appears to be reliable for the very low strain rates
found in the creep test.
The effect of stress on time to rupture can also be correlated with the
timetemperature parameter 6,
Or = tr exp ^^ = F{a) (1322)
where U is the time to rupture. For engineering analysis of creep, the
stress dependence of the time to rupture is often expressed by the empiri
cal equation
tr = aa"" (1323)
An empirical relationship also exists between the rupture life and the
minimum creep rate e^ such that
log tr f C log e^ = iC (1324)
where C and K are constants for a given alloy.
The activation energy for creep can be evaluated from two creep curves
determined with the same applied stress at two different temperatures.
Thus, for equal values of total creep strain obtained at the two temper
atures, the values of d are equal. Therefore,
di = ti exp = $2 = h exp
AHc = R J'^' Inf^ (1325)
J 1 — i 2 tl
137]
Creep and Stress Rupture 353
A rather extensive correlation' of creep and diffusion data for pure metals
shows that the activation energy for hightemperature creep is approxi
mately equal to the activation energy for selfdiffusion (Fig. 139). At
temperatures below about onehalf the melting point A/fc is a function
of temperature, and no correlation exists with the activation energy for
selfdiffusion. The excellent correlation between the activation energies
for creep and selfdiffusion indicates that dislocation climb is the rate
80
/
/
70
^Fe
IV! >
/
octFe
60
/
/
50
Au /
o /
40
30
Mc
c
/m
V
20
In ,
/n
/
o
10
/
/
n
/
10
20
30 40
A// for creep,
50 60
kcal/mole
70 80
90
Fig. 1 39. Correlation between activation energies for hightemperature creep and
selfdiffusion. (J. E. Dorn, in '^ Creep and Recovery,^' p. 274, Aynerican Society for
Metals, Metals Park, Ohio, 1957.)
determining step for hightemperature creep. The fact that a spectrum
of activation energies is observed at lower temperatures indicates that
the creep process is quite complex in this region.
A simple functional relationship between the steadystate creep rate
and the stress does not exist because the effect of stress depends on the
development of the structural changes due to creep deformation. ^ One
of the most reasonable equations relating creep rate with stress and tem
perature is
em = S' exp ^^ exp B'cT (1326)
1 O. D. Sherby, R. L. Orr, and J. E. Dorn, Trans. AIME, vol. 200, pp. 7180, 1954.
2 A. E. Bayce, W. D. Ludemann, L. A. Shepard, and J. E. Dorn, Trans. ASM,
vol. 52, pp. 451468, 1960.
354 Applications to Materials Testing [Chap. 1 3
Avhere S' is a structuresensitive parameter and B' is a constant which is
independent of temperature and structural changes. For low stresses
(below about 5,000 psi) this equation becomes
A f/
6„. = ;S" exp ^jr c^"' (1327)
Another equation which gives good agreement between minimum creep
rate and stress has been suggested by Conrad.^
t,n = iS exp smh
.(T)
(1328)
In Eq. (1328) the structure term S is assumed constant at the minimum
creep rate and the constant <Tc is dependent on temperature.
Weertman^ has derived an equation for the steadystate creep rate
which is similar to Eq. (1327) by using the assumption that dislocation
climb is the ratecontrolling process. The obstacles to dislocation move
ment are considered to be immobile CottrellLomer dislocations (see Sec.
64). In order to escape from these obstacles, the dislocations climb into
adjoining slip planes, forming lowangle boundaries. The activation
energy for this process is the activation energy for selfdiffusion. A
steady state is established between the generation of dislocations and
their annihilation. An alternative dislocation model of steadystate
creep proposed by Mott^ considers that the ratecontrolling process is
the formation of vacancies due to the movement of jogs on screw dis
locations. This dislocation model leads to an equation for the steady
state creep rate similar to Eq. (1312) in which the stress is present as a
cr/r term. The choice of the proper functional relationship between
creep rate and stress is difficult to make from existing experiment and
theory.
1 38. Presentation of Engineering Creep Data
It should be apparent from the previous sections that knowledge of
the hightemperature strength of metals has not advanced to the point
where creep and stressrupture behavior can be reliably predicted for
design purposes on a theoretical basis. There is no other recourse than
to make an intelligent selection of design stresses from the existing data.
Fortunately, a large number of reliable hightemperature strength data
1 H. Conrad, Trans. ASME, ser. D, vol. 81, pp. 617627, 1959.
2 J. Weertman, J. Appl. Phijs., vol. 26, pp. 12131217, 1955.
= N. F. Mott, "NPL Sj'inposium on Creep and Fracture at High Temperature,"
pp. 2124, H. M. Stationery Office, London, 1955.
Sec. 1 38]
Creep and Stress Rupture 355
have been collected and published by the ASTM^ and by the producers
of hightemperature alloys.
A common method of presenting creep data is to plot the logarithm
of the stress against the logarithm of the minimum creep rate (Fig. 1310).
With this type of plot straight lines will frequently be obtained for the
lower temperatures, but discontinuities due to structural instabilities will
often occur at higher temperatures. Values of minimum creep rate lower
^ 6
^ 5
k 4
1
l6256_
■
__
,^,^^
—
TdOO'F
^^
^

^^
—
J400°F _
^
^'
^ —
^ — ■ —
„,■
—
■ —
J500°F _
^_^,,^ — "
.
— '
_,.
^
^^
188 ^
_ ^
MOpjJ.
^^

^'^
— "^
0.01 0.02 0.03 0.05 0.1 0.2 0.3
Rate of creep, %/l,000 hr
0.5
1.0
Fig. 1310. loglog plot of stress vs. minimum creep rate. For 16256 alloy. {Cour
tesy C. L. Clark, Timken Roller Bearing Co.)
than about 0.001 or 0.01 per cent/hr are generally determined by a
standard creep test, while higher values of minimum creep rate are fre
quently determined by a stressrupture test.
In reporting creep data it is common practice to speak of creep strength
or rupture strength. Creep strength is defined as the stress at a given
temperature which produces a minimum creep rate of a certain amount,
usually 0.0001 per cent/hr or 0.001 per cent/hr. Rupture strength refers
to the stress at a given temperature to produce a life to rupture of a
certain amount, usually 100, 1 ,000, or 10,000 hr. A plot of creep strength
or rupture strength against temperature is another common method of
presenting creep data. When relatively shorttime life is an important
' These data are published in the following ASTM Special Technical Publications:
No. 124 (stainless steels), 1952; No. 151 (chromiummolybdenum steels), 1953; No.
160 (superalloys), 1954; No. 180 (carbon steels), 1955; No. 181 (copper and copper
base alloys), 1956; No. 199 (wrought mediumcarbon alloy steels), 1957.
356 Applications to Materials Testing
[Chap. 1 3
design criterion, it is convenient to present the data as a plot of the time
to produce different amounts of total deformation at different stresses
(Fig. 1311). Each curve represents the stress and time at a fixed
temperature to produce a certain per cent of total deformation (sudden
strain plus creep strain). A separate set of curves is required for each
temperature.
In designing missiles and highspeed aircraft data are needed at higher
temperatures and stresses and shorter times than are usually determined
tor creep tests. A common method of presenting these data is by the
use of isochronous stressstrain curves. Creep tests are conducted at
10
Rate of creep, %/I,000 hr
1.0 0.10
0.01
30
25
l20'
o
o
°;15
in
55 10
5
n
^^^v.^%
'Rupture
strength
'^
,^
^^^
' — i£2j^"\
L... Creep streng
'A
_^
_^1£2%__^
C::^.
>
"" — — ■
U.cO%
. 0.10%
1
^
100
1,000
Time, hr
10,000
100,000
Fig. 1311. Deformationtime curves at 1300°F for 16256 alloy. (C. L. Clarh, in
''Utilization of Heat Resistant Alloys," p. 40, American Society for Metals, Metals
Park, Ohio, 1954.)
different stresses for each temperature of interest. From the family of
creep curves the isochronous stressstrain curves are then obtained by
replotting with stress as the ordinate, strain as the abscissa, and time,
usually ranging from about 5 to 60 min, as the parameter for each stress
strain curve.
139. Prediction of Longtime Properties
Frequently hightemperature strength data are needed for conditions
for which there is no experimental information. This is particularly true
of longtime creep and stressrupture data, where it is quite possible to
find that the creep strength to give 1 per cent deformation in 100,000 hr
Sec. 1 39] Creep and Stress Rupture 357
(11 years) is required, although the alloy has been in existence for only
2 years. Obviously, in such situations extrapolation of the data to long
times is the only alternative. Reliable extrapolation of creep and stress
rupture curves to longer times can be made only when it is certain that
no structural changes occur in the region of extrapolation which would
produce a change in the slope of the curve. Since structural changes
generally occur at shorter times for higher temperatures, one way of
checking on this point is to examine the logstresslogrupture life plot
at a temperature several hundred degrees above the required temperature.
For example, if in 1,000 hr no change in slope occurs in the curve at 200°F
above the required temperature, extrapolation of the lower temperature
curve as a straight line to 10,000 hr is probably safe and extrapolation
even to 100,000 hr may be possible. A logical method of extrapolating
stressrupture curves which takes into consideration the changes in slope
due to structural changes has been proposed by Grant and Bucklin.^
Several suggestions of timetemperature parameters for predicting long
time creep or rupture behavior from the results of shorter time tests at
high temperature have been made. The LarsonMiller parameter^ has
the form
(7^ + 460) (C f log t) = constant (1329)
where T = temperature, °F
/ = time, hr
C = a constant with a value between about 10 and 30
In the original derivation of Eq. (1329) on the basis of reactionrate
theory a value of C = 20 was used. While this is a good approximation
when other data are lacking, for best results C should be considered a
material constant which is determined experimentally. Once the proper
constant has been established, then a plot of log stress vs. the Larson
Miller parameter should give a master plot which represents the high
temperature strength of the material for all combinations of temperature
and time. However, since the temperature term in this parameter is
given considerable weight, the LarsonMiller parameter is not very sensi
tive to small changes in rupture life due to structural changes in the
material.
The parameter 6 suggested by Dorn [see Eq. (1317)] is another time
temperature parameter. While this parameter has been useful for cor
relating creep and stressrupture data for pure metals and dilute solid
solution alloys, it has not been used to any great extent with engineering
hightemperature alloys.
An extension of the LarsonMiller parameter has been suggested by
1 N. J. Grant and A. G. Bucklin, Trans. ASM, vol. 42, pp. 720761, 1950.
2 F. R. Larson and J. Miller, Trans. ASME, vol. 74, pp. 765771, 1952.
358 Applications to Materials Testing [Chap. 1 3
Manson and Haferd.^ The MansonHaferd parameter has the form
T  Ta
log t — log ta
= constant
(1330)
where T = test temperature, °F
t = time, hr
Ta, log ta = constants derived from test data
A straight line is obtained when log t is plotted against T for data obtained
100
10
/
r
/•

10
20
/
^4
y*^
.30^
35 " ^
— o
o
/
/
_
40 °
/^
m O)
Tensile (
C=25)
2 S
x<
/
/
o 6 = 0.0108 min~
A f = 0.0318 min"
+ 6 = 0.0868 min~
n CrBf^D[C = P4.P\
I
1
45;:
en
o
+
/
*
Hardness
{C = P5)

50^
100
Hardness, DPH
1,000
Fig. 1312. Hothardnessstrength correlation for iron20 per cent chromium alloy.
{E. E. Underwood, Trans. ASM, vol. 49, p. 403, 1957.)
at a constant stress. The lines obtained at different stresses converge
toward a point with coordinates Ta, log ta.
A comparison of these various parameters with experimental data
shows that the MansonHaferd parameter usually results in the best
overall agreement,^ although in certain cases^ none of these parameters
1 S. S. Manson and A. M. Haferd, NACA Tech. Note 2890, March, 1953; S. S. Man
son, G. Succop, and W. R. Brown, Jr., Trans. ASM, vol. 51, pp. 911934, 1959.
2 R. M. Goldhoff, Trans. ASME, ser. D, vol. 81, pp. 629644, 1959.
3 F. Garofalo, G. V. Smith, and B. W. Royle, Trans. ASME, vol. 78, pp. 14231434,
1956.
Sec. 1310] Creep and Stress Rupture 359
has been found to provide satisfactory correlation of the data. In
general, these parameters provide useful methods for presenting large
numbers of data through the use of master plots of log stress vs. time
temperature parameter. Considerable use has been made of these
parameters for predicting longtime data on the basis of shorttime
results. While there is no assurance that erroneous predictions will
not result from this procedure, it is generally considered that the use of
these parameters for this purpose will probably give better results than
simple graphical extrapolation of loglog plots by one or two logarithmic
cycles.
A shortcut approach to hightemperature properties is through the
use of the hothardness test. There are certain parallelisms between
hightemperature strength results and hothardness data. Just as a
linear relationship exists between stress and the logarithm of the rupture
time, a similar relationship holds between hot hardness and the logarithm
of the indentation time. In Sec. 1110 it was shown that a linear rela
tion exists between hot hardness and tensile strength. Figure 1312 illus
trates a similar relation between hot hardness and hightemperature
strength.^ From the lefthand ordinate we get the relationship between
tensile strength or creep strength and hot hardness. However, each point
on the curve establishes a certain value of the LarsonMiller parameter
along the righthand ordinate. Thus, with a value of hightemperature
stress and a value of the parameter we can establish values of temper
ature and rupture time. This procedure has been verified for both single
phase alloys and complex multiphase hightemperature alloys. It is a
useful method for obtaining hightemperature properties of comparatively
brittle materials. On the other hand, tests of this type give no indication
of the ductility of the material, which in certain cases may be a more
controlling factor than the strength.
1310. Hightemperature Alloys
Hightemperature alloys are a particular class of complex materials
developed for a very specific application. Some appreciation of the
metallurgical principles behind the development of these alloys is impor
tant to an understanding of how metallurgical variables influence creep
behavior. The development of hightemperature alloys has, in the main,
been the result of painstaking, empirical investigation, and it is really
only in retrospect that principles underlying these developments have
become evident.
The nominal compositions of a number of hightemperature alloys are
1 E. E. Underwood, Materials & Methods, vol. 45, pp. 127129, 1957; J. Inst.
Metals, vol. 88, pp. 266271, 19591960.
360 Applications to Materials Testing
[Chap. 1 3
given in Table 131. Only a few of the many available alloys' could be
included in this table. The ferritic alloys were developed first to meet
increased temperature requirements in steam power plants. They are
essentially carbon steels, with increased chromium and molybdenum to
form complex carbides, which resist softening. Molybdenum is particu
larly effective in increasing the creep resistance of steel. Because of
Table 131
Compositions of Typical Hightemperature Alloys
Alloy
C Cr Ni Mo Co W Cb Ti Al
Fe Other
Ferritic Steels
1.25 Cr, Mo
5 Cr, Mo
"1722A" S
410
316
347
16256
A286
Inconel
Inconel X
Nimonic 90
Hastelloy B
Ren^ 41
Udimet 500
Vitallium (HS21;.
X40 (HS31)
N155 (Multimelt)
S590
S816
K42 B
Refractaloy 26 ... .
0.10
0.20
0.30
0.10
1.25
5.00
1.25
12.0
0.50
0.50
0.50
Balance
Balance
Balance
Balance
Austenitic Steels
0.08
0.06
0.10
0.05
17.0
18.0
16.0
15.0
12.0
12.0
25.0
26.0
2.50
6.00
1.25
1
Balance
Balance
Balance
Balance
0.70
1.95
0.20
Nickelbase Alloys
0.04
0.04
0.08
0.10
0.10
0.10
15.5
15.0
20.0
1.0
19.0
19.4
76.0
75.0
58.0
65.0
53.0
55.6
7.0
7.0
0.5
5.0
2.0
0.6
28
10
4
16
2.5
2.3
0.6
1.4
11
14
3.2
2.9
1.6
2.9
Cobaltbase Alloys
0.25
0.40
27.0
25.0
3.0
10.0
5
62
55
1.0
1.0
8
Complex Superalloys
0.15
0.40
0.40
0.05
0.05
21.0
20.0
20.0
18.0
18.0
20.0
20.0
20.0
43.0
37.0
3
4
4
3
20
20
Bal.
22
20
2.5
4.0
4.0
1.0
4.0
4.0
Balance
Balance
3.0
13
18
2.5
2.8
0.2
0.2
0.15A^
1 H. C. Cross and W. F. Simmons, Alloys and Their Properties for Elevated Tem
perature Service, "Utilization of Heat Resistant Alloys," American Society for
Metals, Metals Park, Ohio, 1954.
Sec. 1310] Creep and Stress Rupture 361
oxidation and the instability of the carbide phase these alloys are limited
in use to about 1000°F. Owing to the increased oxidation resistance of
austenitic stainless steels, these alloys extend the useful stressbearing
range to about 1200°F.
The superalloys for jetengine applications are based on either nickel
or cobalt austenitic alloys or combinations of the two. In general the>
contain appreciable chromium for oxidation resistance. Singlephase
solidsolution alloys such as Nichrome (NiCr) and austenitic stainless
steel become weak above about 1300°F. Superalloys, therefore, are
multiphase alloys which attain their strength primarily from a dis
persion of stable secondphase particles. In the cobaltbase alloys and
certain complex NiCoCrFe alloys like N155, S590, and S816 the
secondphase particles are complex metal carbides. Molybdenum, tung
sten, and columbium are added to these alloys to form stable complex
carbides. The carbon content of these alloys is usually critical and must
be controlled carefully. Increasing the carbon content up to a certain
limit increases the amount of carbide particles, and hence the rupture
strength is increased. However, if too much carbon is present, the car
bides will no longer be present as discrete particles. Instead, massive
carbide networks form and reduce the rupture strength. Certain carbide
systems undergo a series of complex aging reactions which may result in
additional strengthening.
Nickelbase superalloys may be strengthened by the addition of small
amounts of Al and/or Ti. The intermetallic compounds NisAl or NiaTi
are formed by these additions. Up to three out of every five atoms of
Al in NisAl may be replaced by Ti, to form Ni3(Al,Ti). Since this com
pound produces greater hardening than NisAl, it is customary to add
both Al and Ti to these nickelbase alloys. Nickelbase alloys contain
ing Al and Ti are true agehardening systems, so that their hightem
perature strength depends on the ability of the system to resist overaging.
The heat treatment must be carefully controlled to put these alloys in the
condition of maximum particle stability. This can be destroyed if the
service conditions fluctuate above the optimum aging temperature.
A new class of dispersionstrengthened hightemperature alloys are
being developed in which thermally stable secondphase particles, chiefly
AI2O3, Si02, and Zr02, are introduced into a metal matrix by artificial
means. ^ The prototype for this development was the sintered alumi
numpowder (SAP) alloy, in which fine AI2O3 particles were dispersed in
an aluminum matrix owing to the breakoff of surface oxide during the
extrusion of sintered aluminum powder. At present most developments
are centered in the preparation of a dispersionhardened alloy by powder
metallurgy methods. Mixtures of fine oxides and metal powders are
» N. J. Grant and O. Preston, Trans. AIMS, vol. 209, pp. 349356, 1957.
362 Applications to Materials Testing
[Chap. 13
pressed, sintered, and extruded into useful shapes. Data for copper and
nickel alloys prepared in this way show that the stressrupture curves
for a dispersionstrengthened alloy drop off much less rapidly with time
than for the same metal without secondphase particles. By comparing
on the basis of rupture time at a given stress, improvements of 1,000 per
cent have been reported for dispersionstrengthened alloys, while increases
in rupture stress at 100 hr of over 10,000 psi have been found for nickel.
300
500
1000
Temperature, °F
1500
2000
Fig. 1313. Typical curves of stress for rupture in 1,000 hr versus temperature for
selected engineering alloys. (Z). P. Moon and W. F. Simmons, DMIC Memo 92,
Battelle Memorial Institute, Mar. 23, 1961.)
The production of artificial dispersionstrengthened alloys offers con
siderable promise, since in principle it permits the use of a secondphase
particle with maximum resistance to growth, and at the same time the
size and amount of the particles can be controlled.
An analogous approach is the development of cermets, in which ceramic
particles such as borides, carbides, and silicides are combined with a
metallic binder by powdermetallurgy methods. Cermets have shown
very high strength at 1800°F, but their use has been limited owing to
poor ductility and insufficient thermal and mechanicalshock resistance.
The hightemperature strength of metals is approximately related to
Sec. 1311] Creep and Stress Rupture 363
their melting points. For example, aluminum has better hightemper
ature strength than zinc, and copper is better than aluminum. Titanium,
chromium, columbium, molybdenum, tantalum, and tungsten, in order
of increasing melting point, are the relatively common metals with melt
ing points greater than iron, cobalt, and nickel, the principal constituents
of presentday superalloys. Although the hightemperature strength of
titanium has not proved to be so great as would be expected on the basis
of its melting point, molybdenum and its alloys have the best hightem
perature strength of any common metals available at the present time.
Figure 1313, which shows the stress for rupture in 1,000 hr at different
temperatures for a number of engineering materials, illustrates the superi
ority of molybdenum. Unfortunately, the use of molybdenum is limited
by the fact that it undergoes catastrophic oxidation at elevated temper
ature. Oxidationresistance coatings for molybdenum are being used to
a certain extent. Extensive development of ways of preparing, purifying,
and fabricating the other refractory metals is under way. This work
includes a search for methods of increasing their oxidation resistance and
hightemperature strength by alloying.
1311. EFfect of Metallursical Variables
The hightemperature creep and stressrupture strengths are usually
higher for coarsegrain material than for a finegrainsize metal. This,
of course, is in contrast to the behavior at lower temperature, where a
decrease in grain size results in an increase in strength. The basic effect
of grain size on hightemperature strength is clouded because it is practi
cally impossible to change the grain size of complex superalloys without
inadvertently changing some other factor such as the carbide spacing or
the aging response. In fact, Parker' has proposed that the observed
dependence of hightemperature strength on grain size is basically incor
rect. Formerly it was suggested that the lower creep strength of fine
grain material was due to greater grainboundary area available for grain
boundary sliding. However, work which indicates that grainboundary
sliding can account for only about 10 per cent of the total creep strain
has cast doubt on this explanation.
Since hightemperature creep depends on dislocation climb, which in
turn depends upon the rate at which vacancies can diffuse to edge dis
locations, Parker feels that the grainsize effect on creep is due to the
effect of grainboundary structure on vacancy diffusion. Vacancy diffu
sion is more rapid along highenergy grain boundaries than through the
bulk lattice. Therefore, with finegrain material with many highangle
grain boundaries dislocation climb will be rapid, and the creep rate is high.
1 E. R. Parker, Trans. ASM, vol. 50, pp. 52104, 1958.
364 Applications to Materials Testing
[Chap. 13
However, when the same material is heated to a high temperature to
coarsen the grain size, most of the highenergy grain boundaries disappear
owing to grain growth. The boundaries which remain are mostly lower
energy grain boundaries, for which vacancy diffusion is relatively slow.
Therefore, dislocation climb will be slower in the coarsegrain material.
The fact that lowangle grain boundaries improve creep properties is
shown in Fig. 1314. Networks of lowangle boundaries were produced
in nickel by straining small amounts in tension at room temperature and
annealing at 800°C. Note the change in shape of the creep curves and
the decreasing creep rate as more and more lowangle grain boundaries
were introduced, with greater prestrain. Recent experiments by Parker
0.10
Fig. 1 31 4. Effect of substructure introduced by prestrain and anneal treatments on
the creep curve of highpurity nickel. {E. R. Parker, Trans. ASM, vol. 50, p. 86,
1958.)
in which the grain size of copper was changed without changing the pro
portion of lowangle and highangle grain boundaries have showai that
the creep rate increases with increasing grain size. Thus, it appears as if
the fundamental dependence of creep rate on grain size is no different
from the dependence of strength on grain size at low temperature. How
ever, secondary effects which accompany the change in grain size when it
is accomplished by commercial annealing procedures are responsible for
the effect which is usually observed.
Comparison of the properties of cast and forged hightemperature
alloys in the same state of heat treatment shows that cast alloys gener
ally have somewhat higher hot strength and creep resistance than forgings
at temperatures above the equicohesive temperature. The reasons for
Sec. 1311] Creep and Stress Rupture 365
this are not well established, but it appears that it is related to a strength
ening from the dendritic structure of the casting. Advantages can be
ascribed to both casting and forging as a method of production for high
temperature alloys. Most hightemperature alloys are hotworked only
with difficulty. There are certain alloys, particularly cobaltbase alloys,
which cannot be hotworked, and therefore they must be produced as
castings. Although castings show somewhat better strength than forg
ings, the production conditions must be carefully controlled to provide
uniform properties. Variations in grain size due to changes in section
size may be a problem with castings. Because of their worked structure,
forgings are more ductile than castings, and generally they show fatigue
properties which are superior to those of castings.
Since hightemperature alloys are designed to resist deformation at
high temperatures, it is not surprising to find that they present problems
in mechanical w^orking and fabrication. Hot working of superalloys for
ingot breakdown is generally done in the range 1700 to 2200°F. It should
be recognized that working highly alloyed materials like superalloys does
not constitute true hot working since residual strains will be left in the
lattice. For this reason the resulting properties of agehardenable alloys
are dependent on the hotworking conditions. The rupture properties
of certain nonaging solidsolution alloys like 16256, N155, and S816
are appreciably improved by controlled amounts of reduction in the range
1200 to 1700°F. This procedure is known as hotcold working, or warm
working. Deformation in this temperature region for these highly alloyed
materials is about equivalent to working mild steel at room temperature.
Warm working and cold working at room temperature of less highly
alloyed materials produce higher creep strength because of the energy
stored in the lattice as a result of the plastic deformation. For a given
operating temperature there will be a critical amount of cold work beyond
which the increased lattice strain causes rapid recovery and recrystalliza
tion. The critical amount of cold work decreases as the operating tem
perature is increased.
In general, there is little correlation between roomtemperature
strength and hightemperature strength. For example, the incorpo
ration of a dispersion of fine oxide particles in the metal matrix may
produce only a modest increase in roomtemperature strength, but the
improvement in rupture time at elevated temperature may be a thousand
fold. The importance of using a thermally stable metallurgical structure
for longtime hightemperature service is well illustrated by the case of
lowalloy steels. Although a quenched and tempered martensitic struc
ture of fine carbides has the best strength at room temperature and may
have good strength for short times up to 1100°F, on long exposure at
elevated temperature the carbides grow and coalesce and the creep
366 Applications to Materials Testing [Chap. 1 3
properties are very poor. Much better creep properties are obtained if
the steel is initially in the stable annealed condition. The problem of
selecting the best heat treatment for a complex superalloy is frequently
difficult. A compromise must be reached between the fineness of the
dispersion of secondphase particles and their thermal stability. Assum
ing that the transformation characteristics for the alloy are completely
known, which is usually not the case, the selection of the heat treatment
would be based on the expected service temperature and the required
service life. One of the difficulties which must be guarded against is the
formation of grainboundary precipitates, which lead to intergranular
fracture.
Hightemperature properties can show considerable scatter, and fre
quently measurably different results are obtained between different heats
of the same material or even between different bars from the same heat.
The creep properties of steels are particularly subject to variations in
properties, which are related in complex ways to the composition, melting
practice, type of mechanical working, and microstructure. Aluminum
deoxidized steels generally have poorer creep properties than silicon deoxi
dized steels. Not only does aluminum refine the grain size, but it also
accelerates spheroidization and graphitization. Considerable improve
ment in the hightemperature properties of superalloys results from
vacuum melting. Fabricability is also improved, presumably because
of the decrease in the number and size of inclusions. In general, life to
rupture and ductility at fracture are both increased by vacuum melting.
Better control of composition, and therefore more uniform response to
heat treatment, is obtained with vacuum melting.
The environment surrounding the specimen can have an important
influence on hightemperature strength. Creep tests on zinc single crys
tals showed that creep practically stopped when copper was plated on
the surface of the specimens.^ When the copper was electrolytically
removed, creep began again at nearly the original rate. Stressrupture
tests on nickel and nickelchromium alloys show a complex dependence
on atmosphere.^ At high temperatures and low strain rates these mate
rials are stronger in air than in vacuum, while the reverse is true at low
temperatures and high strain rates. This behavior is attributed to the
competing effects of strengthening resulting from oxidation and weaken
ing due to lowering of the surface energy by absorbed gases. The nature
of the oxidation can have an important influence on the hightemperature
properties. A thin oxide layer or a finely dispersed oxide will usually
lead to strengthening, but intergranular penetration of oxide will usually
lead to decreased rupture life and intergranular fracture. When mate
1 M. R. Pickus and E. R. Parker, Trans. AIMS, vol. 191, pp. 792796, 1951.
2 P. Shahinian and M. R. Achter, Trans. ASM, vol. 51, pp. 244255, 1959.
Sec. 1 31 3] Creep and Stress Rupture 367
rials must operate in an atmosphere of hot combustion gases or in cor
rosive environments, the service life is materially reduced.
1 31 2. Creep under Combined Stresses
Considerable attention has been given to the problem of design for
combined stress conditions during steadystate creep. ^ In the absence
of metallurgical changes, the basic simplifying assumptions of plasticity
theory (see Sec. 38) hold reasonably well for these conditions. The
assumption of incompressible material leads to the familiar relationship
61 4 «2 + fs = 0. The assumption that principal shearstrain rates are
proportional to principal shear stresses gives
ji^lj2 ^ ^^^^ ^ hj^^ ^ ^ ^^3_3j^
0"! — 0'2 0'2 — (73 (Ti — (T\
Combining these eciuations results in
ei =^kl  K(CT2 + CT3)] (1332)
Similar expressions are obtained for ez and is.
For engineering purposes the stress dependence of the creep rate can be
expressed by Eq. (1316). For combined stress conditions e and a must
be replaced by the effective strain rate e and the effective stress a [see
Eqs. (335) and (336)]. Thus, we can write
i = Ba"' (1333)
Combining Eqs. (1332) and (1333) results in
€1 = Ba"''[ai  H(^2 + C73)] (1334)
The effective stress and the effective strain rate are useful parameters
for correlating steadystate creep data. When plotted on loglog coordi
nates, they give a straightline relationship. Correlation has been
obtained between uniaxial creep tests, creep of thickwalled tubes under
internal pressure, and tubes stressed in biaxial tension.^
1 31 3. Stress Relaxation
Stress relaxation under creep conditions refers to the decrease in stress
at constant deformation. When stress relaxation occurs, the stress
^ The original analysis of this problem was given by C. R. Soderberg, Trans. ASME,
vol. 58, p. 733, 1936. Subsequent analysis has been made by I. Finnie, Trans. ASME,
ser. D, vol. 82, pp. 462464, 1960. For a critical review see A. E. Johnson, Met.
Reviews, vol. 5, pp. 447506, 1960.
2 E. A. Davis, Trans. ASME, ser. D, vol. 82, pp. 453461, 1960.
368 Applications to Materials Testing
[Chap. 1 3
needed to maintain a constant total deformation decreases as a function
of time. Consider a tension specimen which is under a total strain e at
an elevated temperature where creep can occur.
= 6e +
(1335)
where e = total strain
€e = elastic strain
Cp = plastic (creep) strain
For the total strain to remain constant as the material creeps, it is neces
sary for the elastic strain to decrease. This means that the stress required
to maintain the total strain decreases with time as creep increases.
The relaxation of stress in bolted joints and shrink or pressfit assem
blies may lead to loose joints and leakage. Therefore, stressrelaxation
Time
Fig. 1 31 5. Stressrelaxation curves.
tests are commonly made on bolting materials for hightemperature
service.^ Figure 1315 shows the type of curves which are obtained.
The initial rate of decrease of stress is high, but it levels off because the
stress level is decreased and the transient creep rate decreases with time.
These type of curves also can be used to estimate the time required to
relieve residual stress by thermal treatments. If Eq. (1316) can be used
to express the stress dependence of creep, then the time required to relax
the stress from the initial stress at to a is given by^
t =
BE{n'  l)(r"'
1
©"
(1336)
BIBLIOGRAPHY
"Creep and Fracture of Metals at High Temperatures," H. M. Stationery Office,
London, 1956.
^ Relaxation properties of steels and superalloys are given in ASTM Spec. Tech.
PuU. 187, 1956.
2 E. L. Robinson, Proc. ASTM, vol. 48, p. 214, 1948.
Creep and Stress Rupture 369
"Creep and Recoverj^," American Society for Metals, Metals Park, Ohio, 1957.
Finnie, I., and W. R. Heller: "Creep of Engineering Materials," McGrawHill Book
Company, Inc., New York, 1959.
Hehemann, R. F., and G. M. Ault (eds.): "High Temperature Materials," John Wiley
& Sons, Inc., New York, 1959.
"High Temperature Properties of Metals," American Society for Metals, Metals Park,
Ohio, 1951.
Smith, G. v.: "Properties of Metals at Elevated Temperatures," McGrawHill Book
Company, Inc., New York, 1950.
Sully, A. H.: Recent Advances in Knowledge Concerning the Process of Creep in
Metals, "Progress in Metal Physics," vol. 6, Pergamon Press, Ltd., London, 1956.
"Utilization of Heat Resistant Alloys," American Society for Metals, Metals Park,
Ohio, 1954.
Chapter 14
BRITTLE FAILURE
AND IMPACT TESTING
141. The Brittlefailure Problem
During World War II a great deal of attention was directed to the
brittle failure of welded Liberty ships and T2 tankers.^ Some of these
ships broke completely in two, while, in other instances, the fracture did
not completely disable the ship. Most of the failures occurred during
the winter months. Failures occurred both when the ships were in heavy
seas and when they were anchored at dock. These calamities focused
attention on the fact that normally ductile mild steel can become brittle
under certain conditions. A broad research program was undertaken
to find the causes of these failures and to prescribe the remedies for their
future prevention. In addition to research designed to find answers to
a pressing problem, other research was aimed at gaining a better under
standing of the mechanism of brittle fracture and fracture in general.
Many of the results of this basic work are described in Chap. 7, which
should be reviewed before proceeding further with this chapter. While
the brittle failure of ships concentrated great attention on brittle failure
in mild steel, it is important to understand that this is not the only appli
cation where brittle fracture is a problem. Brittle failures in tanks,
pressure vessels, pipelines, and bridges have been documented^ as far back
as the year 1886.
Three basic factors contribute to a brittlecleavage type of fracture.
They are (1) a triaxial state of stress, (2) a low temperature, and (3) a
high strain rate or rapid rate of loading. All three of these factors do
not have to be present at the same time to produce brittle fracture. A
' M. L. Williams, Analysis of Brittle Behavior in Ship Plates, Symposium on Effect
of Temperature on the Brittle Behavior of Metals with Particular Reference to Low
Temperatures, ASTM Spec. Tech. Publ. 158, pp. 1144, 1954.
2 M. E. Shank, A Critical Survey of Brittle Failure in Carbon Plate Steel Structures
Other than Ships, ASTM Spec. Tech. Publ. 158, pp. 45110, 1954.
370
Sec. 142] Brittle Failure and Impact Testing 371
triaxial state of stress, such as exists at a notch, and low temperature are
responsible for most service failures of the brittle type. However, since
these effects are accentuated at a high rate of loading, many types of
impact tests have been used to determine the susceptibility of materials
to brittle fracture. Steels which have identical properties when tested in
tension or torsion at slow strain rates can show pronounced differences in
their tendency for brittle fracture when tested in a notchedimpact test.
However, there are certain disadvantages to this type of test, so that
much work has been devoted to the development of additional tests for
defining the tendency for brittle fracture, and much effort has been
expended in correlating the results of different brittlefracture tests.
Since the ship failures occurred primarily in structures of welded con
struction, it was considered for a time that this method of fabrication
was not suitable for service where brittle fracture might be encountered.
A great deal of research has since demonstrated that welding, per se,
is not inferior in this respect to other types of construction. However,
strict quality control is needed to prevent weld defects which can act as
stress raisers or notches. New electrodes have been developed that make
it possible to make a weld with better properties than the mildsteel plate.
The design of a welded structure is more critical than the design of an
equivalent riveted structure, and much effort has gone into the develop
ment of safe designs for welded structures. It is important to eliminate
all stress raisers and to avoid making the structure too rigid. To this
end, riveted sections, known as crack arresters, were incorporated in some
of the wartime ships so that, if a brittle failure did occur, it would not
propagate completely through the structure.
142. Notchedbar Impact Tests
Various types of notchedbar impact tests are used to determine the
tendency of a material to behave in a brittle manner. This type of test
will detect differences between materials which are not observable in a
tension test. The results obtained from notchedbar tests are not readily
expressed in terms of design requirements, since it is not possible to meas
ure the components of the triaxial stress condition at the notch. Further
more, there is no general agreement on the interpretation or significance
of results obtained with this type of test.
A large number of notchedbar test specimens of different design
have been used by investigators of the brittle fracture of metals. Two
classes of specimens have been standardized^ for notchedimpact testing.
Charpy bar specimens are used most commonly in the United States,
while the Izod specimen is favored in Great Britain. The Charpy speci
1 ASTM Standards, pt. 3, 1958, Designation E2356T,
372 Applications to Materials Testing
[Chap. 14
men has a square cross section and contains a notch at the center of its
length. Either a V notch or a keyhole notch is used. The Charpy
specimen is supported as a beam in a horizontal position. The load is
applied by the impact of a heavy swinging pendulum (approximately
16 ft /sec impact velocity) applied at the midspan of the beam on the side
opposite from the notch. The specimen is forced to bend and fracture
at a strain rate on the order of 10* in. /(in.) (sec). The Izod specimen is
either circular or square in cross section and contains a V notch near one
end. The specimen is clamped vertically at one end like a cantilever
beam and is struck with the pendulum at the opposite end. Figure 141
illustrates the type of loading used with these tests. Note that the notch
is subjected to a tensile stress as the specimen is bent by the moving
pendulum. Plastic constraint at the notch produces a triaxial state of
stress similar to that shown in Fig. 710. The relative values of the three
Impact
load
7777777,
Impact
load
7/7777/
Charpy Vnotch Izod
Fig. 141. Sketch showing method of loading in Charpy and Izod impact tests.
principal stresses depend strongly on the dimensions of the bar and the
details of the notch. For this reason it is important to use standard
specimens. The value of the transverse stress at the base of the notch
depends chiefly on the relationship between the width of the bar and the
notch radius. The wider the bar in relation to the radius of the notch,
the greater the transverse stress.
The response of a specimen to the impact test is usually measured by
the energy absorbed in fracturing the specimen. For metals this is usu
ally expressed in footpounds and is read directly from a calibrated dial
on the impact tester. In Europe impact results are frequently expressed
in energy absorbed per unit crosssectional area of the specimen. Very
often a measure of ductility, such as the per cent contraction at the
notch, is used to supplement this information. It is also important to
examine the fracture surface to determine whether it is fibrous (shear
failure) or granular (cleavage fracture). Figure 142 illustrates the
appearance of these two types of fractures.
The notchedbar impact test is most meaningful when conducted over
a range of temperature so that the temperature at which the ductileto
)ec.
142]
Brittle Failure and Impact Testing 373
4:60
o 40
.S 20
Fig. 142. Fracture surfaces of Charpy specimens tested at different temperatures.
Left, 40°F; center, 100°F; right, 212°F. Note gradual decrease in the granular region
and increase in lateral contraction at the notch with increasing temperature.
brittle transition takes place can be determined. Figure 143 illustrates
the type of curves which are obtained. Note that the energy absorbed
decreases with decreasing temperature but that for most cases the
decrease does not occur sharply at
a certain temperature. This makes
it difficult to determine accurately
the transition temperature. In
selecting a material from the stand
point of notch toughness or tend
ency for brittle failure the im
portant factor is the transition
temperature. Figure 143 illus
trates how reliance on impact
resistance at only one temperature
can be misleading. Steel A shows
higher notch toughness at room
temperature; yet its transition
temperature is higher than that of
steel B. The material with the lowest transition temperature is to be
preferred.
Notchedbar impact tests are subject to considerable scatter,^ particu
larly in the region of the transition temperature. Most of this scatter is
1 R. H. Frazier, J. W. Spretnak, and F. W. Boulger, Symposium on Effect of Tem
perature on the Brittle Behavior of Metals, ASTM Spec. Tech. Publ. 158, pp. 286307,
1954.
80
40 +40
Temperature. °F
+80
Fig. 143. Transitiontemperature curves
for two steels, showing fallacy of depend
ing on roomtemperature results.
374 Applications to Materials Testing
[Chap. 14
due to local variations in the properties of the steel, while some is due to
difficulties in preparing perfectly reproducible notches. Both notch
shape and depth are critical variables, as is the proper placement of the
specimen in the impact machine.
The shape of the transition curve depends on the type of test and also
on the material. For example, keyhole Charpy specimens usually give
60
50
^ 40
r
=^30
a>
i5 20
10
100
80
fe"60
to
40
20
Semikilled mild steel  0.1 8 percent C
0.54 percent Mn
0.07 percent Si
Energy transition
Charpy
keyhole
60 40 20 20 40 60
Ductility transition
100 120 140
20 40 60
Temporcture, "F
100 120 140 160
Fig. 7 44. Transitiontemperature curves based on energy absorbed, fracture appear
ance, and notch ductility. {.W ■ S. Pellini, ASTM Spec. Tech. Publ. 158, p. 222,
1954.)
Sec. 143] Brittle Failure and Impact Testing 375
a sharper breaking curve than Vnotch Charpy specimens. For a tough
steel Vnotch Charpy specimens generally give somewhat higher values
than keyhole specimens. The transition temperature for a given steel
will be different for differentshaped specimens and for different types
of loading with different states of stress. The correlation of transition
temperatures measured in different ways will be discussed in a later
section.
Because the transition temperature is not sharply defined, it is impor
tant to understand the criteria which have been adopted for its definition.
The most suitable criterion for selecting the transition temperature is
whether or not it correlates with service performance. In general, cri
teria for determining the transition temperature are based on a transition
in energy absorbed, change in the appearance of the fracture, or a tran
sition in the ductility, as measured by the contraction at the root of the
notch.' Figure 144 shows that the same type of curve is obtained for
each criterion. This figure also illustrates the relative shapes of the
curves obtained with keyhole and Vnotch Charpy specimens. The
energy transition temperature for Vnotch Charpy specimens is fre
quently set at a level of 10 or 15 ftlb. Where the fracture appearance
changes gradually from shear through mixtures of shear and cleavage
to complete cleavage, with decreasing temperature, the transition tem
perature is frequently selected to correspond to a temperature where
50 per cent fibrous (shear) fracture is obtained. The ductility transition
temperature is sometimes arbitrarily set at 1 per cent lateral contraction
at the notch. One characteristic of these criteria is that a transition
temperature based on fracture appearance always occurs at a higher tem
perature than if based on a ductility or energy criterion.
143. Slowbend Tests
The slow bending of flatbeam specimens in a testing machine is some
times used as a method of determining the transition temperature. A
biaxial state of stress is produced during the bending of an unnotched
beam when the width is much greater than the thickness. When the
ratio of width to thickness is close to unity, the stress is essentially uni
axial, but as the width increases, the ratio of the transverse to longi
tudinal stress approaches a value of }'2, the condition for a state of plane
stress.^ The unnotchedbend test represents a condition of severity inter
' W. S. Pellini, Evaluation of the Significance of Charpy Tests, Symposium on
Effect of Temperature on the Brittle Behavior of Metals with Particular Reference
to Low Temperatures, ASTM Spec. Tech. Publ. 158, pp. 216261, 1954.
2 G. S. Sangdahl, E. L. Aul, and G. Sachs, Proc. Soc. Exptl. Stress Anal., vol. 6,
no. 1, pp. 118, 1948.
376 Applications to Materials Testing
[Chap. 14
mediate between that of the tensile test and a notchedimpact test. Usu
ally a notch is used to introduce triaxial stress, in which case the transition
temperature is raised.
The effect of adding the variable of high strain rate is complex. In a
comparison of the transition temperature measured with a slowbend test
and a Charpy impact test with identically notched specimens it was found
that the ductility transition was raised by impact but that the fracture
transition was lower for the impact test. From this and other work, it
appears as if the fracture transition temperature is not sensitive to
strain rate.
45° notch O.Ol" radius 0.05"deep
Weld bead. /^ 4"
KINZEL BEND SPECIMEN
Notch radius 1 mm  0.080" deep
LEHIGH. BEND SPECIMEN
Fis. 1 45. Notchbend test specimens.
The Kinzel and Lehigh (Fig. 145) notchbend specimens are frequently
used for studying the effect of welding and metallurgical variables on
notch toughness. Both specimens incorporate a longitudinal weld bead
which is notched so that the weld metal, the heataffected zone, and the
unaffected base metal are exposed to the stress at the root of the notch.
Both specimens are bent with the load applied opposite to the notch.
The Lehigh specimen provides duplicate tests. Loaddeflection curves
are obtained and the data plotted in terms of energy absorbed to maxi
mum load, energy absorbed after maximum load to fracture, or total
energy absorbed. Lateral contraction at the notch and bend angle are
also measured.
Sec. 1 44]
Brittle Failure and impact Testing 377
144. Specialized Tests for Transition Temperature
A number of new tests for determining the transition temperature of
steel have been developed as a result of the research on the brittle failure
of ships. Space will permit only a brief description of several of the most
interesting of these tests, which give indication of attaining more general
acceptance.
A number of tests subject the notch to simultaneous tension and
bending. This can be done by eccentrically loading a notched tensile
Sow cut
Impact ^ (^ A_
Liquid Ng
coolant
Weld
Heat
applied
[a)
Fig. 146. (a) Specimen used in Navy tear test; (6) specimen used in Robertson test.
specimen or by using a specimen such as shown in Fig. 146a. This
specimen is used in the Navy tear test.^ It employs the full thickness
of the steel plate. The advantage of a combined tension plus bending
load over one of bending alone is that by suitably increasing the tensile
load the compression region developed by the bending load can be elimi
nated. Since a highcompression region will retard crack propagation, a
test which combines both bending and tension aids in crack propagation.
Robertson^ devised an interesting test for determining the temperature
at which a rapidly moving crack comes to rest. A uniform tensile stress
1 N. A. Kahn and E. A. Imbemo, Welding J., vol. 29, pp. 153s165s, 1949.
2 T. S. Robertson, Engineering, vol. 172, pp. 445448, 1951; J. Iron Steel Inst.
{London), vol. 175, p. 361, 1953.
378 Applications to Materials Testing [Chap. 14
is applied to a specimen of the type shown in Fig. 1466. A starter crack
is cut with a jeweler's saw at one side of the specimen. This side is
cooled with liquid nitrogen, and the other side is kept at a higher tem
perature. Thus, a temperature gradient is maintained across the width
of the specimen. A crack is started at the cold end by an impact gun.
The energy available from the impact is not sufficient to make the crack
grow very large, but the applied tensile stress tends to keep it growing.
The crack travels across the width of the specimen until it reaches a
point where the temperature is high enough to permit enough yielding
to stop the crack. This occurs when the plastic deformation required
for further spread of the crack cannot be supplied by the stored elastic
energy. For each applied tensile stress there is a temperature above
which the crack will not propagate. Robertson's data showed that this
arrest temperature decreases sharply for most mildsteel plates when the
applied tensile stress is lowered to about 10,000 psi. For these steels
the crack will not be arrested if the stress exceeds this value and the
temperature is below room temperature. This test has been modified
for use without a temperature gradient.^
The dropweight test was developed by the Naval Research Labo
ratory^ to measure susceptibility to the initiation of brittle fracture in
the presence of a cracklike notch. The specimen is a flat plate with a
3in.long bead of hardfacing metal applied at the center and notched to
half depth. The welded side of the specimen is placed face down over
end supports, and the center of the specimen is struck with a 60lb falling
weight. The bead of hardfacing metal cracks in a brittle manner, pro
ducing a sharp, cracklike notch. Since the purpose of the dropweight
test is to see whether or not fracture will occur at a sharp notch when
the amount of yielding that can occur is restricted, the bending fixture is
designed so as to limit the deflection of the specimen to 5°. Only 3° of
bend is needed to produce a crack in the brittle weld bead. The addi
tional 2° of bend provides a test of whether or not the steel can deform
in the presence of the cracklike notch. This is a "gono go" type of test
in that at a given temperature the specimen either fractures completely or
remains intact. The highest temperature of fracture is termed the nil
ductility transition temperature. This test provides a sharp transition
temperature and is quite reproducible.
The explosionbulge test was developed by the Naval Research Labo
ratory^ to measure susceptibility to propagation of brittle fracture. A
crackstarter weld is applied to the center of a 14in.square plate. The
1 F. J. Feely, D. Hrtko, S. R. Kleppe, and M. S. Northrup, Welding J., vol. 33,
pp. 99sllls, 1954.
2 Pellini, op. cit., pp. 233235.
3 Ibid., pp. 228231.
145]
Brittle Failure and Impact Testing 379
specimen is placed over a circular die and subjected to the force of a
controlled explosion. The explosion produces a compressive shock wave
which is reflected from the bottom of the plate as a tensile wave. This
test is interpreted in terms of the appearance of the fracture in the plate.
At a higher temperature the plate bulges, but the cracks still run to the
edges of the plate. At still higher temperatures the plate bulges con
siderably more, and the crack becomes a shear crack which is confined to
the center of the specimen. The fractureappearance transition temper
ature is selected as the temperature at which cracking is confined to the
bulged region of the plate. For most steels this transition temperature
will fall 40 to 60°F above the nilductility transition of the dropweight
test. The two tests supplement each other. The dropweight test
establishes a temperature below which the material is very susceptible
to fracture initiation, while the explosionbulge test establishes a temper
ature above which the material is immune to brittlefracture propagation.
145. SisniFicance of the Transition Temperature
The notch toughness of a material should really be considered in
terms of two distinct transition temperatures. Figure 147 shows the
transitiontemperature curves for
t
Ductility
transition
tennperature
such an ideal material. The duc
tility transition temperature is re
lated to the fractureinitiation
tendencies of the material. Com
pletely brittle cleavage fracture
occurs readily below the ductility
transition temperature. The frac
tureappearance transition temper
ature is related to the crackprop
agation characteristics of the
material. Above the fracture
transition temperature cracks do
not propagate catastrophically,
because fracture occurs by the
shear mode with appreciable ab
sorption of energy. In the region between these two transition temper
atures fractures are difficult to initiate, but once initiated they propa
gate rapidly with little energy absorption.
Actual materials do not have two distinct transition temperatures
such as were shown in Fig. 147. Instead, Fig. 148 is more character
istic of the type of curves that are obtained with Charpy Vnotch tests
on mild steel. The ductility transition temperature usually occurs at
Temperature — >
Fig. 147. Concept of two transition
temperatures.
380 Applications to Materials Testing
[Chap. 14
an energy level of 5 to 20 ftlb. Frequently a value of 15 ftlb is used to
establish this transition temperature. The ductility transition temper
ature may also be determined from measurements of the contraction at
the root of the notch. The fractureappearance transition temperature
is measured by the per cent shear in the fracture surface. Usually it is
taken at the temperature at which 50 per cent fibrous fracture is obtained.
The fracture transition temperature always occurs at a higher temper
ature than the ductility transition temperature. For a given material
the fractureappearance transition temperature is fairly constant regard
less of specimen geometry, notch sharpness, and rate of loading. On the
other hand, the ductility transition temperature depends very strongly
on the testing conditions. The ductility transition temperature is usu
Ductility
transition
Shear fracture
Ductile
behavior
in service
Difficult crack
initiation and
propogation
Temperature — ^
Fig. 148. Significance of regions of transitiontemperature curve.
ally more pertinent to service performance, because if it is difficult to
initiate a crack, then it is not necessary to worry about its propagation.
There is no general correlation between any of the brittlefracture tests
and service performance. The greatest number of data exist for failed
hull plates in welded ships. Tests on these steels showed that they all
had Charpy Vnotch values of 11.4 ftlb or less when tested at the
temperature at which failure occurred. Experience with rimmed and
semikilled mildsteel plates in thicknesses up to 1 in. indicates that a
minimum Charpy Vnotch value of 15 ftlb at the lowest operating tem
perature should prevent brittle fracture if the nominal stresses are of the
order of onehalf the yield point. For higher alloy steels a higher value
of minimum impact resistance may be required.
Part of the difficulty in correlating notchimpact data has been caused
by failure to recognize and distinguish between the two general types
Sec. 146] Brittle Failure and Impact Testing 381
of transitiontemperature criteria. Comparisons should not be made
between test results where the two criteria have been mixed. For exam
ple, no correlation is found between the keyhole and Vnotch Charpy
tests when the transition temperature is measured at a level correspond
ing to 50 per cent of the maximum energy. This is because the 50 per
cent energy level is close to the ductility transition for the keyhole speci
men but near the fracture transition for the Vnotch specimen. Good
correlation is found between the two specimens when they are both evalu
ated with common ductility criteria. Further, good correlation has been
obtained' between the nilductility transition measured by the drop
weight test and the Charpy Vnotch test. Good correlation has been
shown between the dropweighttest nilductility transition and service
fractures. The correlation problem is well illustrated by the work of the
Ship Structure Committee.^
146. Metallurgical Factors Affecting Transition Temperature
Changes in transition temperature of over 100°F can be produced by
changes in the chemical composition or microstructure of mild steel.
The largest changes in transition temperature result from changes in the
amount of carbon and manganese.^ The 15ftlb transition temperature
for Vnotch Charpy specimens (ductility transition) is raised about 25°F
for each increase of 0.1 per cent carbon. This transition temperature is
lowered about 10°F for each increase of 0.1 per cent manganese. Increas
ing the carbon content also has a pronounced effect on the maximum
energy and the shape of the energy transitiontemperature curves (Fig.
149). The Mn:C ratio should be at least 3:1 for satisfactory notch
toughness. A maximum decrease of about 100°F in transition temper
ature appears possible by going to higher Mn : C ratios. The practical
limitations to extending this beyond 7 : 1 are that manganese contents
above about 1.4 per cent lead to trouble with retained austenite, while
about 0.2 per cent carbon is needed to maintain the required tensile
properties.
Phosphorus also has a strong effect in raising the transition temper
ature. The 15ftlb Vnotch Charpy transition temperature is raised
about 13°F for each 0.01 per cent phosphorus. Since it is necessary to
control phosphorus, it is not generally advisable to use steel made by the
Bessemer process for lowtemperature applications. The role of nitrogen
is difficult to assess because of its interaction with other elements. It is,
1 H. Greenberg, Metal Progr., vol. 71, pp. 7581, June, 1957.
2 E. R. Parker, "Brittle Behavior of Engineering Structures," chap. 6, John Wiley
& Sons, Inc., New York, 1957.
3 J. A. Rinebolt and W. J. Harris. Jr.. Trans. ASM, vol. 43, pp. 11751214, 1951.
382 Applications to Materials Testing
[Chap. 14
however, generally considered to be detrimental to notch toughness.
Nickel is generally accepted to be beneficial to notch toughness in
amounts up to 2 per cent and seems to be particularly effective in lower
ing the ductility transition temperature. Silicon, in amounts over
0.25 per cent, appears to raise the transition temperature. Molybdenum
raises the transition almost as rapidly as carbon, while chromium has
little effect.
Notch toughness is particularly influenced by oxygen. For highpurity
iron^ it was found that oxygen contents above 0.003 per cent produced
intergranular fracture and corresponding low energy absorption. When
the oxygen content was raised from 0.001 per cent to the high value of
280
Corbon
■200
200
Temperature, °F
400
Fig. 149. Effect of carbon content on the energytransitiontemperature curves for
steel. {J. A. Rinebolt and W. J. Harris, Jr., Trans. ASM, vol. 43, p. 1197, 1951.)
0.057 per cent, the transition temperature was raised from 5 to 650°F.
In view of these results, it is not surprising that deoxidation practice has
an important effect on the transition temperature. Rimmed steel, with
its high iron oxide content, generally shows a transition temperature
above room temperature. Semikilled steels, which are deoxidized with
silicon, have a lower transition temperature, while for steels which are
fully killed with silicon plus aluminum the 15 ftlb transition temper
ature will be around — 75°F. Aluminum also has the beneficial effect of
combining with nitrogen to form insoluble aluminum nitride. The use
of a fully killed deoxidation practice is not a completely practical answer
to the problem of making steel plate with high notch toughness because
there is only limited capacity for this type of production.
Grain size has a strong effect on transition temperature. An increase
1 W. P. Rees, B. E. Hopkins, and H. R. Tipler, /. Iron Steel Inst. (London), vol. 172
pp. 403409, 1952.
Sec. 146] Brittle Failure and Impact Testing 383
of one ASTM number in the ferrite grain size (actually a decrease in
grain diameter) can result in a decrease in transition temperature of
30°F for mild steel. Decreasing the grain diameter from ASTM grain
size 5 to ASTM grain size 10 can change the 10ftlb Charpy Vnotch
transition temperature^ from about 70 to — 60°F. A similar effect of
decreasing transition temperature with decreasing austenite grain size is
observed with higher alloyed heattreated steels. Many of the variables
concerned with processing mild steel affect the ferrite grain size and
therefore affect the transition temperature. Since normalizing after hot
rolling results in a grain refinement, if not carried out at too high a tem
perature, this treatment results in reduced transition temperature. The
cooling rate from the normalizing treatment and the deoxidation practice
are variables which also must be considered. Air cooling and aluminum
deoxidation result in a lower transition temperature. Using the lowest
possible finishing temperature for hot rolling of plate is also beneficial.
For a given chemical composition and deoxidation practice, the tran
sition temperature will be appreciably higher in thick hotrolled plates
than in thin plates. This is due to the difficulty of obtaining uniformly
fine pearlite and grain size in a thick section. Generally speaking, allow
ance for this effect must be made in plates greater than ^ in. in thickness.
The notch toughness of steel is greatly influenced by microstructure.
The best notch toughness is obtained with a microstructure which is
completely tempered martensite. A completely pearlitic structure has
poor notch toughness, and a structure which is predominately bainite
is intermediate between these two. As an example of the effect of micro
structure on transition temperature, in an SAE 4340 steel for which a
tempered martensitic structure and pearlitic structure were compared at
the same hardness, it was found that the Charpykeyhole transition tem
perature at 25 ftlb was 350°F lower for the tempered martensitic struc
ture. Further discussion of the notch toughness of heattreated steels
will be found in Sec. 148.
Lowcarbon steels can exhibit two types of aging phenomena which
produce an increase in transition temperature. Quench aging is caused
by carbide precipitation in a lowcarbon steel which has been quenched
from around 1300°F. Strain aging occurs in lowcarbon steel which has
been coldworked. Cold working by itself will increase the transition
temperature, but strain aging results in a greater increase, usually around
40 to 60°F. Quench aging results in less loss of impact properties than
strain aging. The phenomenon of blue hrittleness, in which a decrease in
impact resistance occurs on heating to around 400°F, is due to strain
aging.
1 W. S. Owen, D. H. Whitmore, M. Cohen, and B. L. Averbach, Welding J., vol. 36,
pp. 503s511s, 1957.
384 Applications to Materials Testing
[Chap. 14
The notchedimpact properties of rolled or forged products vary
with orientation in the plate or bar. Figure 1410 shows the typical
form of the energytemperature curves for specimens cut in the longi
tudinal and transverse direction of a rolled plate. Specimens A and B
are oriented in the longitudinal direction of the plate. In specimen A
the notch is perpendicular to the plate, while in B the notch lies parallel
to the plate surface. The orientation of the notch in specimen A is
generally preferred. In specimen C this notch orientation is used, but
the specimen is oriented transverse to the rolling direction. Transverse
specimens are used in cases where the stress distribution is such that the
crack would propagate parallel to the rolling direction. Reference to
80
:? 60
S40
l5 20
+40
Temperature, °F
+ 80
+120
Fig. 1410. Effect of specimen orientation on Charpy transitiontemperature curves.
Fig. 1410 shows that quite large differences can be expected for different
specimen orientations at high energy levels, but the differences become
much less at energy levels below 20 ftlb. Since ductility transition tem
peratures are evaluated in this region of energy, it seems that specimen
and notch orientation are not a very important variable for this criterion.
If, however, materials are compared on the basis of roomtemperature
impact properties, orientation can greatly affect the results.
147. Effect of Section S
ize
Difficulty with brittle fracture usually increases as the size of the struc
ture increases. This is due to both metallurgical and geometrical factors.
Sec. 148] Brittle Failure and Impact Testing 385
In the previous section it was shown that the transition temperature of a
given steel usually decreases with increasing plate thickness because of
the increased grain size produced in hotrolling thick plates. However,
Charpy tests on specimens of varying size but identical metallurgical
structure and geometrically similar notches show that there is a size
effect. At some temperature the largest specimens will be completely
brittle, while the small specimens will be completely ductile. The frac
tures for inbetween specimens will vary from almost fully ductile to
almost fully brittle.
A dramatic demonstration of size effect was obtained in tests of ship
hatch corners carried out at the University of California. Full scale,
onehalf scale, and onequarter scale models were tested. These models
were similar in all details and were made from the same material by the
same welding procedures. When fracture strength was measured in
terms of pounds per square inch of the net crosssectional area, the full
sized specimen had only about onehalf the strength of the quarterscale
model.
The higher transition temperature or lower fracture stress of large
structures is due to two factors. The larger structure can contain a
more unfavorable state of stress due to stress raisers, and it also pro
vides a large reservoir for stored elastic energy. Since the Griffith cri
terion requires that the elastic strain energy must provide the surface
energy for the formation of the fracture surface, the greater the available
stored energy, the easier it is for the attainment of an uncontrollable,
rapidly spreading crack.
148. Notch Toughness of Heattreated Steels
It has been demonstrated many times that a tempered martensitic
structure produces the best combination of strength and impact resist
ance of any microstructure that can be produced in steel. In Chap. 9
it was shown that the tensile properties of tempered martensites of the
same hardness and carbon content are alike, irrespective of the amount
of other alloy additions. This generalization holds approximately for
the roomtemperature impact resistance of heattreated steels, but it is
not valid for the variation of impact resistance with temperature. Fig
ure 1411 shows the temperature dependence of impact resistance for a
number of different alloy steels, all having about 0.4 per cent carbon and
all with a tempered martensite structure produced by quenching and
tempering to a hardness of Re 35. Note that a maximum variation of
about 200°F in the transition temperature at the 20ftlb level is possible.
Even greater spread in transition temperature would be obtained if the
tempering temperature were adjusted to give a higher hardness.^ Slack
1 H. J. French, Trans. AIME, vol. 206, pp. 770782, 1956.
386
Applications to Materials Testing
[Chap. 14
quenching so that the microstructure consists of a mixture of tempered
martensite, bainite, and peariite results in even greater differences between
alloy steels and in a general increase in the transition temperature.
The energy absorbed in the impact test of an alloy steel at a given
test temperature generally increases with increasing tempering temper
ature. However, there is a minimum in the curve in the general region
80
70
60
50
S40
°30
20
10
4^
^
Q
uenc
nart
hed
ensit
to
, , /ir\
r
e
^
^
^
A
"
/
^
^
/
/
/
1
5145
/
/
^
^340
t64C
/
/
/
^
o 2 yj c:
^i
/
^
'
/
1
/
/
h
fl
\.
/
/
/
A
V
J^
4
V
/
r
=^
^
^
_>
V
X
/
^3
\^
y
300 200
100 100
Test temperature.
200
300
400
Fig. 1411. Temperature dependence of impact resistance for different alloy steels
of same carbon content, quenched and tempered to Re 35. {H. J. French, Trans.
AIMS, vol. 206, p. 770, 1956.)
of 400 to 600°r (Fig. 1412). This has been called 500°F emhrittlement,
but because the temperature at which it occurs depends on both the
composition of the steel and the tempering time, a more appropriate
name is temperedmartensite emhrittlement. Emhrittlement of steel in this
tempering region is one of the chief deterrents to using steels at strength
levels much above 200,000 psi. Studies of this emhrittlement phenome
non have shown' that it is due to the precipitation of platelets of cementite
iL. J. Klingler, W. J. Barnett, R. P. Frohmberg, and A. R. Troiano, Trans. ASM,
vol. 46, pp. 15571598, 1954.
149]
Brittle Failure and Impact Testing 387
from ecarbide during the second stage of tempering. These platelets
have no effect on the reduction of area of a tensile specimen, but they
severely reduce the impact resistance. They can be formed at temper
atures as low as 212°F and as high as 800°F, depending on the time
allowed for the reaction. Silicon additions of around 2.25 per cent are
50
45
40
35
30
25
4330 8 4340: Ch
Klingler, Bornett,
orpy
Frohm
V notch
berg,
and Troiano 1
2340 a 4140: Charpy keyholenotch
/
1
bros
smann
y
/
/
'l
^
^
i
/
^
V.
\^
■
/
^
/
^i^
/
^
1^
^

^
200
400 600 800
Tempering temperature, °F
1000
1200
Fig. 1 41 2. Effect of tempering temperature on impact resistance at room temperature
for four alloy steels quenched to martensite. {fl. J. French, Trans. AIMS, vol. 206,
p. 770, 1956.)
effective in increasing the temperature at which the platelets precipitate,
and this permits tempering in the region 500 to 600°F, without severe
embrittlement.
149. Temper Embrittlement
Temper embrittlement^ refers to a loss in notch toughness of plain
carbon and alloy steels when exposed at temperatures above about 700°E"
but below the temperature for the formation of austenite. Alloy steels
are particularly susceptible to embrittlement when they are tempered in
the region 800 to 1100°F or slowcooled through this temperature range.
1 The extensive literature in this field has been reviewed by B. C. Woodfine, J.
Iron Steel Inst. (London), vol. 173, pp. 229240, 1953, and L. D. Jaffe, Welding J.,
vol. 34, pp. 14121502, 1955.
388 Applications to Materials Testing [Chap. 14
This can become a particularly important problem with heavy sections
that cannot be cooled through this region rapidly enough to suppress
embrittlement. Temper embrittlement also can be produced by iso
thermal treatments in this temperature region. The kinetics of the
process produces a Cshaped curve when some parameter of embrittle
ment is plotted on temperaturetime coordinates. More rapid embrittle
ment results from slow cooling through the critical temperature region
than from isothermal treatment. Temper enibrittlement can be com
pletely eliminated from an embrittled steel by heating into the austenite
region and cooling rapidly through the embrittling temperature region.
The presence of temper embrittlement is usually determined by meas
uring the transition temperature by means of a notchedbar impact test.
The hardness and tensile properties are not sensitive to the embrittle
ment, except for very extreme cases, but the transition temperature can
be increased around 200°F by ordinary embrittling heat treatments. The
fracture of a temperembrittled steel is intergranular, while the brittle
fracture of a nonembrittled steel is transgranular. This suggests that
temper brittleness is due to a grainboundary weakness. However, no
evidence for a grainboundary film or precipitate has been uncovered
from studies by means of the electron microscope of the microstructure
of temperembrittled steel. Therefore it is generally hypothesized that
temper embrittlement is due to the segregation of impurities to the grain
boundaries without the formation of an observable precipitate phase.
The effect of various alloying elements on this embrittlement can then be
explained on the basis of their rates of diffusion and relative solubilities
at the grain boundaries and within the grains. Much more information
is needed before a detailed mechanism of temper embrittlement can be
determined.
Molybdenum is the only alloying element which decreases the suscepti
bility to temper embrittlement. The best solution to the problem is to
avoid tempering in the region of greatest susceptibility to embrittlement.
Tempering at a higher temperature for a short time may be better than
a long tempering treatment at a lower temperature. A water quench
from the tempering temperature will serve to minimize embrittlement
on cooling.
1410. Hydroscn Embrittlement
Severe embrittlement can be produced in many metals by very small
amounts of hydrogen. Bodycentered cubic and hexagonal closepacked
metals are most susceptible to hydrogen embrittlement. As little as
0.0001 weight per cent of hydrogen can cause cracking in steel. Face
centered cubic metals are not generally susceptible to hydrogen embrittle
1410]
Brittle Failure and Impact Testing 389
ment.^ Hydrogen may be introduced during melting and entrapped
during solidification, or it may be picked up during heat treatment,
electroplating, acid pickling, or welding.
The chief characteristics of hydrogen embrittlement are its strainrate
sensitivity, its temperature dependence, and its susceptibility to delayed
fracture. Unlike most embrittling phenomena, hydrogen embrittlement
is enhanced by slow strain rates. At low^ temperatures and high tem
peratures hydrogen embrittlement is negligible, but it is most severe in
some intermediate temperature region. For steel the region of greatest
susceptibility to hydrogen embrittlement is in the vicinity of room tem
perature. Slow bend tests and notched and unnotched tension tests will
detect hydrogen embrittlement by a drastic decrease in ductility, but
notchedimpact tests are of no use for detecting the phenomenon.
A common method of studying hydrogen embrittlement is to charge
notched tensile specimens with know^n amounts of hydrogen, load them
to different stresses in a deadweight machine, and observe the time to
failure. A typical delayedfracture curve is shown in Fig. 1413. Note
that the notched tensile strength of
a charged specimen may be much
lower than the strength of a hydro
genfree specimen. There is a re
gion in which the time to fracture
depends only slightly on the applied
stress. There is also a minimum
critical value below which delayed
fracture wnll not occur. The simi
larity of the delayed fracture curve
to the fatigue SN curve has led to
the use of the term "static fatigue"
for the delayedfracture phenome
non. The minimum critical stress,
or "static fatigue limit," increases with a decrease in hydrogen content
or a decrease in the severity of the notch. The hydrogen content of
steel may be reduced by "baking," or heating at around 300 to 500°F.
Hydrogen is present in solution as monatomic hj^drogen. Because it is
a small interstitial atom, it diffuses very rapidly at temperatures above
room temperatvire. A commonly held concept of hydrogen embrittle
ment is that monatomic hydrogen precipitates at internal voids as
molecular hydrogen. These voids might be true voids, microcracks, or
perhaps simply regions of high dislocation density. As hydrogen diffuses
1 The familiar example of the embrittlement of copper by hydrogen at elevated
temperature is due to the reaction of hydrogen with oxygen to form internal pockets
of steam.
/!/o Ay
.ZL^f^gen
o200
o
o_
^
]00
. A
"q.
x^arged with hydrogen
<
1 1 1 1 1
0.01
0.1 1 10 100
Time to fracture, hr
1,000
Fis. 1413. Delayedfracture curve.
390 Applications to Materials Testing [Chap. 14
into the voids, the pressure builds up and produces fracture. While this
concept explains the general idea of hydrogen embrittlement, it is not in
agreement with all the experimental facts. Further insight into the
mechanism has resulted from the work of Troiano and coworkers.^ By
determining the rate of crack propagation by means of resistivity meas
urements they were able to show that the crack propagates discontinu
ously. This indicates that the rate of crack propagation is controlled by
the diffusion of hydrogen to the region of high triaxial stress just ahead
of the crack tip. When a critical hydrogen concentration is obtained,
a small crack forms and grows, to link up with the previous main crack.
The fact that the time for the initiation of the first crack has been found
to be insensitive to applied stress supports the idea that the proc
ess depends on the attainment of a critical hydrogen concentration.
The main effect of stress is to assist in the accumulation of this concen
tration. The minimum critical stress in Fig. 1413 can be interpreted
as the stress needed to cause a critical accumulation of hydrogen. The
higher the average hydrogen content, the lower the necessary critical
stress.
The formation of hairline cracks, or flakes, in large ingots and forgings
during cooling or roomtemperature aging has long been attributed to the
presence of hydrogen. Studies^ of flake formation have shown that in
addition to containing hydrogen the steel must be subjected to transfor
mation stresses for flaking to occur. The hydrogen content necessary
for flaking varies widely with composition, size, and segregation. Flakes
have been observed in steel with as low as 3 ppm of hydrogen. On the
other hand, very high amounts of hydrogen can be tolerated without
causing flakes if the transformation stresses are minimized by decom
posing the austenite above the Ms temperature before cooling to room
temperature.
1411. Flow and Fracture under Very Rapid Rates of Loading
The mechanical properties of metals can be appreciably changed when
they are subjected to very rapidly applied loads. Shock loading can be
produced by highvelocity impact machines^ or by shock waves from the
detonation of explosives.* In considering dynamic loading of this type
it is important to consider effects due to stresswave propagation within
1 A. R. Troiano, Trans. ASM, vol. 52, pp. 5480, 1960.
2 A. W. Dana, F. J. Shortsleeve, and A. R. Troiano, Trans. AIME, vol. 203, pp.
895905, 1955.
3 P. E. Duwez and D. S. Clark, Proc. ASTM, vol. 47, pp. 502532, 1947.
^ J. S. Rinehart and J. Pearson, "Behavior of Metals under Impulsive Loads,"
American Society for Metals, Metals Park, Ohio, 1954.
Sec. 1411] Brittle Failure and Impact Testing 391
the metal.' This is because a rapidly applied load is not instantaneously
transmitted to all parts of the loaded body. Rather, at a brief instant
after the load has been applied the remote portions of the body remain
undisturbed. The deformation and stress produced by the load move
through the body in the form of a wave that travels with a velocity ot
the order of several thousand feet per second. Compression waves are
generated in a metal when it is subjected to an explosive blast (impulsive
loading), while tensile waves can be produced by a tensionimpac\
machine. The propagation velocity of a compressive or tensile stress
wave is given by
c. = ('^' (141)
where Co = velocity of wave propagation
da/de = slope of stressstrain curve
p = density of metal
If the wave amplitude is low, so that the elastic limit is not exceeded,
Eq. (141) can be written
Co = (T (142)
Corresponding to the wave velocity co, a certain particle velocity Vp is
produced in the metal. The wave velocity and particle velocity are in
the same direction for a compressive wave, but they are in opposite direc
tions for a tensile wave. The particle velocity Vp is related to the wave
velocity co by the following equations:
Vp = f\,de = ~ ("— (143)
Jo P Jo Co
These equations can be used to determine the stress or strain in a dynami
cally loaded metal provided that the wave and particle velocities can be
determined. When a bar is subjected to tension impact, it is found that
there is a critical velocity which produces rupture at the impacted end
at the instant of impact. By combining Eqs. (141) and (143) the
equation for the critical velocity is obtained.
where €„ is the strain corresponding to the tensile strength of the metal.
For shock loads below Vu the bar would undergo deformation but would
not fracture. The value of critical impact velocity for most metals lies
in the range of 200 to 500 ft/sec.
1 H. Kolsky, "Stress Waves in Solids," Oxford University Press, New York, 1953
392 Applications to Materials Testing [Chap. 1 4
Measurement of the dynamic stressstrain curve is difficult because of
the short time during which events occur and because care must be taken
to consider all wavepropagation phenomena. The information which is
available indicates that for shock loading the stressstrain curve is raised
about 10 to 20 per cent compared with the static curve. There is gener
ally an increase in the energy to fracture with increasing impact velocity
up to the point where the critical velocity is reached.
Marked differences occur between fracture under impulsive loads and
under static loads. With impulsive loads there is not time for the stress
to be disturbed throughout the entire body, so that fracture can occur in
one part of the body independently of what happens in another part.
The velocity of propagation of stress waves in solids lies in the range
3,000 to 20,000 ft/sec, while the velocity of crack propagation is about
6,000 ft/sec. Therefore, with impulsive loads it may be found that
cracks have formed but did not have time to propagate before the stress
state changed. Reflections of stress waves occur at free surfaces and
fixed ends, at changes in cross section, and at discontinuities within the
metal. A compression wave is reflected from a surface as a tension wave,
and it is this reflected tension wave which in most cases causes fracture
under impulsive loading. When a thick plate is subjected to explosive
loading against one surface, the interference from the incident and
reflected wave from the opposite surface will cause a tensile stress to be
built up a short distance from the opposite surface. The tensile stress
may be high enough to cause fracture, and the plate is said to have
scabbed. From studying^ the thickness of the scabs it is possible to
arrive at values for a critical normal fracture stress.
BIBLIOGRAPHY
Parker, E. R.: "Brittle Behavior of Engineering Structures," John Wiley & Sons
Inc., New York, 1957.
Queneau, B. R. : "The Embrittlement of Metals," American Society for Metals,
Metals Park, Ohio, 1956.
Shank, M. E.: "Control of Steel Construction to Avoid Brittle Failure," Welding
Research Council, New York, 1957.
Symposium on Effect of Temperature on the Brittle Behavior of Metals with Par
ticular Reference to Low Temperatures, ASTM Spec. Tech. Publ. 158, 1954.
Tipper, C. F.: The Brittle Fracture of Metals at Atmospheric and Subzero Temper
atures, Met. Reviews, vol. 2, no. 7, pp. 195261, 1957.
1 J. S. Rinehart, J. Appl. Phys., vol. 22, p. 555, 1951; On Fractures Caused by
Explosions and Impacts, Quart. Colo. School Mines, vol. 55, no. 4, October, 1960.
Chapter 15
RESIDUAL STRESSES
1 51 . Origin of Residual Stresses
Residual stresses are the system of stresses which can exist in a body
when it is free from external forces. They are sometimes referred to as
internal stresses, or lockedin stresses. Residual stresses are produced
whenever a body undergoes nonuniform plastic deformation. For exam
ple, consider a metal sheet which is being rolled under conditions such
that plastic flow occurs only near the surfaces of the sheet (Fig. 15la).
(a) id)
Fig. 151. (a) Inhomogeneous deformation in rolling of sheet; (6) resulting distribu
tion of longitudinal residual stress over thickness of sheet (schematic).
The surface fibers of the sheet are coldworked and tend to elongate,
while the center of the sheet is unchanged. Since the sheet must remain
a continuous whole, the surface and center of the sheet must undergo a
strain accommodation. The center fibers tend to restrain the surface
fibers from elongating, while the surface fibers seek to stretch the central
fibers of the sheet The result is a residualstress pattern in. the sheet
393
394 Applications to Materials Testing [Chap. 15
which consists of a high compressive stress at the surface and a tensile
residual stress at the center of the sheet (Fig. 1516). In general, the
sign of the residual stress which is produced by inhomogeneous defor
mation will be opposite to the sign of the plastic strain which produced
the residual stress. Thus, for the case of the rolled sheet, the surface
fibers which were elongated in the longitudinal direction by rolling are
left in a state of compressive residual stress when the external load is
removed.
The residualstress system existing in a body must be in static equi
librium. Thus, the total force acting on any plane through the body
and the total moment of forces on any plane must be zero. For the
longitudinal stress pattern in Fig. 1516 this means that the area under
the curve subjected to compressive residual stresses must balance the
area subjected to tensile residual stresses. The situation is not quite so
simple as is pictured in Fig. 151. Actually, for a complete analysis, the
residual stresses acting across the width and thickness of the sheet should
be considered, and the state of residual stress at any point is a combined
stress derived from the residual stresses in the three principal directions.
Frequently, because of symmetry, only the residual stress in one direc
tion need be considered. A complete determination of the state of resid
ual stress in three dimensions is a very considerable undertaking.
Residual stresses are to be considered as only elastic stresses. The
maximum value which the residual stress can reach is the elastic limit
of the material. A stress in excess of this value, with no external force
to oppose it, will relieve itself by plastic deformation until it reaches the
value of the yield stress.
It is important to distinguish between macro residual stresses and
microstresses. Macro residual stresses, with which this chapter is pri
marily concerned, vary continuously through the volume of the body
and act over regions which are large compared with atomic dimensions.
Microstresses, or textural stresses, act over dimensions as small as several
unit cells, although their effects may extend throughout most of a grain.
Because of the anisotropy of the elastic properties of crystals, micro
stresses will vary greatly from grain to grain. The back stress developed
by a pileup of dislocations is an example of this type of residual stress.
Another example is the precipitation of secondphase particles from solid
"solution. If the precipitate particles occupy a larger volume than the
;omponents from which they formed, i.e., if the secondphase particles
have a lower density than the matrix, then each particle in t'^ying to
occupy a larger volume is compressed by the matrix. This, in turn,
develops tensile stresses in the matrix in directions radial and tangential
to the secondphase particles. The experimental determination of these
localized stresses in twophase systems is very difficult, although measure
Sec. 151] Residual Stresses 395
ments of their average value have been made with X rays.''^ Calcu
lations of the microstresses existing in twophase systems have been made
by Lazlo/ who uses the terminology "tessellated stresses" for this type
of residual stress. The determination of the microstresses which exist in
plastically deformed singlephase metals is necessary for an understand
ing of the mechanism of strain hardening. Estimates of these micro
stresses can be made from detailed analysis of the broadening of Xray
diffraction lines. Further improvements on the techniques are needed
before these measurements can be used without ambiguity.
Residual stresses arise from nonuniform plastic deformation of a body.
The principal methods by which this can occur are by inhomogeneous
changes in volume and in shape. A third source of residual stress may
exist in builtup assemblies, such as welded structures. Even though
the structure is not subjected to external loads, different members of the
structure may be under stress due to various interactions between the
members of the assembly. This type of residual stress is called reaction
stress. Because it falls in the area of structural engineering, it will not
be considered further in this chapter.
The precipitation of secondphase particles in a metal matrix is an
example of a nonuniform volume change which produces very localized
micro residual stresses. However, if the reaction does not proceed uni
formly over the cross section of the body because of differences in either
chemical composition or rate of heat transfer, there will be a variation
in the distribution of microstresses which will produce macro residual
stresses. Nitriding and carburizing are processes in which a microstress
distribution is produced around each nitride or carbide particle, but
because these diffusioncontrolled reactions occur only on the surface,
there is a nonuniform volume increase in this region. Thus, a macro
compressive residual stress is produced on the surface, and this is bal
anced by tensile residual stresses in the interior. The phase transfor
mation from austenite to martensite which occurs during the quenching
of steel is an outstanding example of a nonuniform volume change lead
ing to high residual stresses. Because of the technological importance
of this situation, it will be considered in a separate section of this chapter.
Volume changes need not necessarily involve rapid quenching or phase
changes to produce residual stresses. In the cooling of a large, hot ingot
of a metal w^hich shows no phase change, the temperature differences
which are present between the surface and the center may be enough to
1 J. Gurland, Trans. ASM, vol. 50, pp. 10631071, 1958.
2 C. J. Newton and H. C. Vacher, J. Research Natl. Bur. Standards, vol. 59, pp. 239
243, 1957.
" For a review of Lazlo's exten:sivo and detailed work, see F. R. N. Nabarro, "Sym
posiuni on Internal Stresses," p. Gl, In.titute of .^!etals, London, 1948.
396 Applications to Materials Testing
[Chap. 1 5
develop residual stresses.^ The edges of a hot slab cool faster than the
center. The thermal contraction of the cooler edges produces a strain
mismatch between the edges and center of the ingot which results in the
distribution of longitudinal stresses shown in Fig. 1526. Since the hot
center has a lower yield stress, it cannot support the compressive stress
imposed on that region and because of plastic deformation the center of
the ingot shrinks to relieve some of the stress (Fig. 152c). When the
center of the slab finally cools, the total contraction will be greater for
the center than the edges because the center contracts owing to both
cooling and plastic deformation (Fig. l52d). The center will then be
stressed in residual tension, and the edges will be in compression.
ia)
(d)
ic)
Fig. 152. Development of residual stresses during cooling of a hot ingot. Cool
portions shown shaded. {After W. M. Baldwin, Jr., Proc. ASTM, vol. 49, p. 541,
1949.)
The forming operations required to convert metals to finished and
semifinished shapes rarely produce homogeneous deformation of the
metal. Each particular plastic forming process has a residualstress
distribution which is characteristic of the process and is influenced to a
certain extent by the way in which the process has been carried out.
Earlier in this section we have seen how the residual stress in the rolling
direction of a sheet results from inhomogeneous deformation through the
thickness of the sheet.
Other metalworking operations which are not generally classed as
metalforming processes can produce residual stresses because they
involve inhomogeneous deformation. Spot welding and butt welding
both produce high tensile stresses at the center of the area of application
of heat. Shot peening, surface hammering, and surface rolling produce
shallow biaxial compressive stresses on a surface which are balanced by
biaxial tension stresses in the interior. As was noted in Chap. 12, shot
peening is an effective method of reducing fatigue failures. Residual
1 W. M. Baldwin, Jr., Proc. ASTM, vol. 49, pp. 539583. 1949.
)ec.
1 52] Residual Stresses 397
stresses are developed in electroplated coatings. Soft coatings such as
lead, cadmium, and zinc creep sufficiently at room temperature to relieve
most of tfiese plating stresses. Hard coatings like chromium and nickel
can have either high tensile or compressive residual stresses, depending
upon the conditions of the plating process.
The superposition of several deformation operations does not produce
a final residualstress distribution which is the algebraic sum of the stress
distributions produced by the preceding operations. In general, the final
deformation process determines the resulting residualstress pattern.
However, superposition of stress distributions is a valid procedure when
one is considering the effect of residual stress on the response of a body
to an external stress system. For all practical purposes residual stresses
can be considered the same as ordinary applied stresses. Thus, a com
pressive residual stress will reduce the effectiveness of an applied tensile
stress in producing fatigue failure, and residual tensile stresses will
increase the ease with which failure occurs.
1 52. Effects of Residual Stresses
The presence of residual stresses can influence the reaction of a mate
rial to externally applied stresses. In the tension test, that region of a
specimen containing high residual tensile stresses will yield plastically at
a lower value of applied stress than a specimen without residual stresses.
Conversely, compressive residual stresses will increase the yield stress.
This fact is used to strengthen gun tubes and pressure vessels by the
process known as autofrettage. In autofrettage thickwalled cylinders are
purposely strained beyond the elastic limit of the material at the bore of
the cylinder so that this region will contain compressive residual stresses
when the cylinder is unloaded. In a relatively brittle material like high
strength steel the presence of tensile residual stresses can cause a decrease
in the fracture strength. The possibility of unknown residual stresses
existing at the root of machined notches is a problem in notch tensile
testing. The effect of residual stresses on fatigue performance is a well
recognized phenomenon and was considered in Sec. 129.
Residual stresses are responsible for warping and dimensional insta
bility. If part of a body containing residual stresses is machined away,
as in machining a long keyway in a colddrawn bar, the residual stresses
in the material removed are also eliminated. This upsets the static equi
librium of internal forces and moments, and the body distorts to establish
a new equilibrium condition. Warping due to redistribution of residual
stresses when surface layers are removed can be exceedingly troublesome,
particularly with precision parts like tools and dies. However, there is
a useful aspect to this, for as will be seen in the next section, the measure
398 Applications to Materials Testing [Chap. 15
ment of dimensional changes when material is removed from a body is
one of the established methods for measuring residual stresses. Dimen
sional instability refers to changes in dimensions which occur without
any removal of material. These changes result from the deformation
required to maintain equilibrium when the residualstress distribution
changes from stress relaxation on longtime roomtemperature aging.
The residualstress pattern in steels may also be altered by the transfor
mation of retained austenite to martensite on aging.
Stress corrosion cracking^ is a type of failure which occurs when cer
tain metals are subjected to stress in specific chemical enviroments.
Residual stress is just as effective as stress due to an externally applied
load in producing stress corrosion cracking. Examples of combinations
which produce stress corrosion cracking are mercury or ammonia com
pounds with brass (season cracking) and chlorides with austenitic stain
less steels and certain agehardenable aluminum alloys. Extreme care
should be taken to minimize residual stress when these situations are
likely to be encountered. In fact, accelerated stress corrosion cracking
may be used as a qualitative test to indicate the presence of residual
stresses. Typical solutions which are used for this purpose are listed
below:
1. Brassmercurous nitrate in water; standardized for detection of
residual stress in brass cartridge cases (ASTM B154)
2. Austenitic stainless steelboiling solution of 10 per cent H2SO4 and
10 per cent CUSO4, or boiling MgCl2
3. Mild steelboiling NaOH
4. AluminumNaCl solution
5. Magnesiumpotassium chromate solution
1 53. Mechanical Methods for Residualstress Measurement
Residual stresses cannot be determined directly from straingage
measurements, as is the case for stresses due to externally applied loads.
Rather, residual stresses are calculated from the measurements of strain
that are obtained when the body is sectioned and the lockedin residual
stresses are released.
The method developed by Bauer and Heyn for measuring the longi
tudinal residual stresses in a cylinder is a good illustration of the tech
niques involved. The residual stresses in the cylinder can be likened to
a system of springs (Fig. 153). In this example the cylindrical bar is
assumed to contain tensile residual stresses around the periphery and
^A review of this important subject is given by W. D. Robertson (ed.), "Stress
Corrosion Cracking and Embrittlement," John Wiley & Sons, Inc., New York, 1956.
2 E. Heyn and O. Bauer, Intern. Z. MetaUog., vol. 1, pp. 1650, 1911.
Sec. 153]
Residual Stresses
399
compressive stresses at the center. By the spring analogy, the center
springs would be compressed and the outer springs elongated (Fig. 15.3a).
Now, if the static equilibrium of forces is upset by removing the outer
springs, the compressed springs will elon
gate (Fig. 1536). The amount of elon ^— 1.
gation experienced by the center springs
is directly proportional to the force ex
erted on them by the outer springs.
The strain experienced by the core
is dei = dLi/L, where Li is the expanded
length of the element. The stress re
lieved by this expansion, of, is related
to the strain through Hooke's law.
(Tc = E dei
Since the cylinder was initially in equi
librium before the skin was removed,
the force in the center core must
balance the force in the removed mate
rial.
[a)
[d)
AiE dei = Pskin
Fig. 1 53. (a) Heyn's spring model
for longitudinal residual stresses in
a cylinder; (b) elongation of center
portion, due to removal of restraint
of outer springs.
If Ao is the original area of the cylin
drical bar, then the area of the skin is
dAi = Aq — Ai. The average stress existing in the skin is as, so that
the force in the skin may be written
Pskin = asdAi
Equating the force in the core and the skin results in an equation for the
average stress in the skin.
AiE dei
dAi
(151)
The above equation expresses the residual stress when it has the very
arbitrary distribution shown in Fig. 154a. Actually, the distribution
of longitudinal residual stress is more likely to vary in the continuous
manner shown in Fig. 1546.
The residualstress distribution shown in Fig. 1546 can be determined
by the Bauer and Heyn method if the stresses are determined by remov
ing thin layers and measuring the deformation in the remaining portion.
If sufficiently thin layers are removed and the process is repeated enough
times, the measured stress distribution will approach the distribution
shown in Fig. 1546. When the stress distribution is measured by the
400
Applications to Materials Testing
[Chap. 15
successive removal of thin layers, Eq. (151) gives the residual stress in
the first layer removed. However, this equation will not give a true
indication of the actual stress which
originally existed in the second radial
layer of the bar, because the removal
of the first layer from, the bar has
caused a redistribution of stress in
the remainder of the bar. The
actual stress in the second layer as it
existed in the original bar is given by
AiE de2
dA2
o
tr
1—
t
^
7/
I
^^//////^
o
O)
Q.
F
> " ia) (Z>)
Fig. 154. Variation of longitudinal
residual stress over the diameter of a
bar. (a) Arbitrary case with constant
residual stress; (b) realistic case with
continuously varying stress from sur
face to center.
02 =
 E dei (152)
where A2 is the area of the cylinder
remaining after layer dAi has been
removed. In determining the stress
in succeeding layers a correction
must be made for the stress relief due to all previous layer removals.
AnE den
dAn
E{dei + de2 + dez +
+ deni)
(153)
If layers of differential thickness are removed, Eq. (153) may be written
de
= E
Equation (154) may be used to determine the longitudinal residual stress
at any radial position in the bar. The best procedure is to plot the axial
strain e after each layer removal against the area of the remaining bar, A .
Connecting the points for successive layers will give a smooth curve.
For any radial position, given by A, this curve will give the value of e,
and the slope of the curve at the point e, A, will be de/dA.
The Bauer and Heyn method has been considered in considerable detail
because it is a simple illustration of the methods used to convert measure
ments of strain into residual stresses. However, this technique can give
values which are considerably in error because it does not consider tan
gential or radial residual stresses. In general, a bar will contain residual
stresses in all three principal directions. The presence of transverse or
radial stresses can result in up to 30 per cent error in the determination
of the longitudinal residual stresses.
Sachs Boringout Method
An exact method of determining the longitudinal, tangential, and radial
residual stresses in bars or tubes was proposed by Mesnager^ and modified
1 M. Mesnager, Compt. rend., vol. 169, pp. 13911393, 1919.
Sec. 153]
Residual Stresses
401
by Sachs. ^ This method is commonly known as the Sachs boringout
technique. The method is Hmited to cylindrical bodies in which the
residual stresses vary in the radial direction but are constant in the
longitudinal and circumferential directions. This is not a particularly
restrictive condition since bars and tubes made by most forming oper
ations have the required symmetrical residualstress pattern.
In using this technique with a solid bar the first step is to drill an
axial hole. Then a boring bar is used to remove layers from the inside
diameter of the hollow cylinder, extreme care being taken to prevent
overheating. About 5 per cent of crosssectional area should be removed
between each measurement of strain. To eliminate end effects, the
specimen length should be at least three times the diameter. After each
layer is removed from the bore, measurements are made of longitudinal
strain cl and tangential strain ei.
eL
et
Li — Lq
Di  Do
Do
The changes in length L and diameter D may be measured with microme
ters, but better accuracy can be obtained by mounting SR4 strain gages^'^
in the longitudinal and circumferential directions of the bar.
In accordance with the Sachs analysis, the longitudinal and tangential
strains are combined in two parameters.
A = cl + vet
6 = e< + veL
The longitudinal, tangential, and radial stresses can then be expressed by
the following equations,
at = E'
<Jr = E'
{Ao  A)
clA
iA.A)f^
A
Ao + A
2A
e
(155)
(^i'e)
» G. Sachs, Z. Metallk., vol. 19, pp. 352357, 1927.
^ J. J. Lynch, "Residual Stress Measurements," pp. 5152, American Society for
Metals, Metals Park, Ohio, 1952.
' A critical evaluation of experimental procedures has been presented by R. A.
Dodd, Metallurgia, vol. 45, pp. 109114, 1952.
402 Applications to Materials Testing
[Chap. 1 5
where E' = E/(l  v"")
Ao = original area of cylinder
A = area of boredout portion of cylinder
V = Poisson's ratio
In using the above equations it is convenient to plot the strain parame
ters and A as functions of the boredout area A. The slopes of these
curves are then used in the above equations. To estimate the stress
along the axis of the bar or at the bore of a tube, it is necessary to extrapo
late the A versus A and 9 versus A curves to ^ =0. Similarly, to deter
mine the stresses on the outer surface of the bar, these curves should be
extrapolated to Jl = ^o There is a limit to how closely the boredout
diameter can approach the outside diameter of the bar without buckling.
Since the residual stress near the outer surface may be rapidly changing
with radial distance an extrapolation in this region may be in error. One
way to get a better estimate of the stresses near the outside surface of the
bar is to measure the changes in diameter of the axial hole while metal is
removed from the outer surface of the bar. The Sachs equations for this
case are given below,
(TL = E'
E'
CFt
(A^.)^A
(A  An)
de A + Ah
dA
2A
e
(156)
= E
'(^'^)
where Ah = area of boredout hole
A = area of cylinder after each layer removal
A method for accurately determining both the longitudinal and tangential
(circumferential) residual stresses in thinwalled tubing has been described
by Sachs and Espey.^
TreutingRead Method
Treuting and Read^ developed a method for determining the biaxial
residualstress state on the surface of a thin sheet. The method assumes
that the metal behaves in an elastically homogeneous manner and that
the stress varies, not in the plane of the sheet, but only through the
thickness. To apply the method, the sheet specimen is cemented to a
flat parallel surface, and the thickness is reduced a certain amount by
careful polishing and etching. The sheet specimen is then released from
1 G. Sachs and G. Espey, Trans. AIMS, vol. 147, pp. 7488, 1942; see also Lynch,
op. cit., pp. 8792.
2 R. G. Treuting and W. T. Read, J. Appl. Phys., vol. 22, pp. 130134, 1951.
>ec.
154]
Residual Stresses
403
the surface, and measurements are made of the longitudinal radius of
curvature Rx, the transverse radius of curvature Ry and the thickness t.
Figure 155 illustrates the orientation of the principal stresses and the
curvature of the sheet.
The measured values of radius of curvature are expressed in terms of
two parameters Px and Py.
p. = ^ + ^
Rx Rv
Py = — + —
Ry Rx
Measurements of Rx and Ry are made for different amounts of metal
removal, and Px and Py are plotted against the sheet thickness t. The
f
/
/a>
}
\ ., J
r
^0
(a)
{b)
Fig. 155. (a) Coordinate system for measuring biaxial stress in thin sheet; (6) curva
ture produced by removing material from top surface.
residual stresses in the x and y directions of the sheet are determined for
any value of t by the following equations:
Cu — —
E
6(1
E
v')
6(1

v'A
(/o + /)2 ^ + 4(<o + t)Px + 2 r P. dt
dP
{u + ty^ + A{t, + t)Py + 2
Pydt
(157)
Values of dP/dt are obtained from the slope of the curves of P versus t,
and the integrals are evaluated by determining the area under the P
versus t curve over the appropriate limits.
1 54. DeHection Methods
Complete analysis of the residual stresses in the principal directions
and their variation with depth in the body by the methods described in
the previous section is a laborous procedure. It is not uncommon to
404 Applications to Materials Testing
[Chap. 1 5
find that nearly 40 hr is required for the complete analysis of a single
specimen. Therefore, approximate methods which are more rapid but
less accurate have been developed. Since the techniques involve a
mechanical slitting of the specimen and measurement of the deflection
of the slit element, they are usually called deflection methods. Deflection
methods can be applied when it is reasonable to assume that the stress
varies linearly through the thickness of a plate or tube but is constant
along the length, width, or circumference. Actually, the stress distri
^'
28
"Residual stress
distribution
(a)
^^'
2S +2r
Fig. 1 56. Determination of longitudinal residual stress by deflection method, (a)
Rolled sheet; (Jo) drawn rod.
bution through the thickness is rarely linear. The formulas that are
presented below give the residual stress only at the surface of the body
and consider the stress only in one particular direction. Although this
is usually the direction of the maximum stress, it should be realized that
the presence of stresses in the other principal directions can affect its
value.  ,
Rolled Sheet
The longitudinal residual stress at the surface of a rolled sheet can be
determined by splitting the sheet down its central plane to release the
bending moment which existed in the body' (Fig. 156a). The bending
^ Lynch, op. at., pp. 8182.
Sec. 154] Residual Stresses 405
moment in this sheet may be expressed as follows,
M = ^ (158)
where E' = E/(l  p')
I = moment of inertia of split section
R = radius of curvature
Although the residualstress distribution which went to make up this
bending moment is unknown, it is assumed for the analysis that it varies
linearly over the thickness t. Based on this linear relationship, the maxi
mum longitudinal stress at the surface is given by the familiar equation
from strength of materials,
^^  n^o^
(^L = J (159)
where c is the distance from the neutral axis to the outer fiber, in this
case t/4. Substituting Eq. (158) into (159) and introducing the value
for c results in
<TL = ^ (1510)
The radius of curvature may be expressed in terms of the deflection 8
and the length of the curved beam L by Eq. (1511) if the deflection is
small compared with the radius of curvature.
K = g . (1511)
Thus, the longitudinal residual stress at the surface is given by
aL = ^, (1512)
Round Bar
The same procedure can be used to estimate the surface longitudinal
stress in a round bar> (Fig. 1566). With the notation given in Fig. 1566
the procedure outlined above results in the following equation:
1.65^^5r
(TL = y^ (1513)
Thinwall Tubing
The longitudinal stress at the surface of a tube may be determined^ by
splitting a longitudinal tongue from the wall of the tube (Fig. 157a).
1 Ibid., p. 82.
* R. J. Anderson and E. G. Fahlman, J. Inst. Metals, vol. 32, pp. 3G7383, 1924.
406 Applications to Materials Testing
[Chap. 1 5
With the notation given in Fig. 157 a, the longitudinal stress is given by
the following equation,
(TL = TIT (1514)
where
E'
E
1  v'
Equation (1514) is the same as Eq. (1512) for a rolled sheet, except that
t/2 is used in the latter case because only half the sheet thickness deflects.
U)
id)
Fig. 157. Determination of residual stresses in thinwall tube by deflection method,
(a) Longitudinal stress; (b) circumferential stress.
Experience with the slitting of tubes has shown that the observed deflec
tion is a function of the width of the tongue. The maximum deflection
is obtained when the tongue width is 0.1 to 0.2 times the diameter of
the tube.
To determine the circumferential residual stresses in a tube, a longi
tudinal slit is made the entire length of the tube, and the change in
Sec. 155] Residual Stresses 407
diameter is measured^ (Fig. 1576). With the notation in the figure, the
circumferential residual stress at the surface is given by
'''\k~wj
1 55. Xray Determination of Residual Stress
The measurement of residual stress with X rays utilizes the interatomic
spacing of certain lattice planes as the gage length for measuring strain.
In essence, the interatomic spacing for a given lattice plane is determined
for the stressfree condition and for the same material containing residual
stress. The change in lattice spacing can be related to residual stress.
Because X rays penetrate less than 0.001 in. into the surface of a metal,
Xray methods measure only surface strains and therefore only surface
residual stresses can be determined. Since no stress exists normal to a
free surface, this method is limited to uniaxial and biaxial states of stress.
For many applications, particularly where fatigue failure is involved, this
is not a serious disadvantage.
Xray methods have the important advantage that they are non
destructive and involve no cutting or slitting of the object to measure
the stresses. Further, it is not always necessary to make measurements
on the specimen in the unstressed condition. This is often an advantage
in making postmortem examinations of parts which failed in service.
The Xray method measures the residual stress in a very localized area,^
since the Xray beam covers an area approximately 3^ in. in diameter.
This makes the Xray technique useful for measurements of sharply
changing stress gradients, but it may be a disadvantage where the objec
tive is to characterize the overall residualstress condition in a surface.
Since the Xray method is based essentially on an accurate measurement
of the shift in position of the Xray reflection from a given set of lattice
planes due to the presence of elastic strain, it is necessary that the dif
fraction lines be accurately located. Using film techniques to record the
Xray reflections requires that the specimen yield sharp diffraction lines
if the lattice strain is to be measured with precision. Since severely
coldworked material and quenched and tempered steel give broad dif
fraction lines, the residual stresses in these important classes of specimens
cannot be accurately determined by Xray film techniques. However,
the introduction of Geigercounter Xray spectrogoniometer equipment
1 D. K. Crampton, Trans. AIME, vol. 89, pp. 233255, 1930.
2 Mechanical methods of measuring localized residual stresses around a small
drilled hole have been developed. See J. Mathar, Trans. ASME, vol. 86, pp. 249
254, 1934.
408 Applications to Materials Testing [Chap. 1 5
has permitted a more accurate measurement of the Xray diffractionline
profile than is possible with film techniques. Methods have been devel
oped of measuring with considerable precision residual stresses in heat
treated steel by means of X rays.^
Bragg's law expresses the relationship between the distance between a
given set of lattice planes, d, and the wavelength of Xray radiation, X,
the order of the diffraction n, and the measured diffraction angle 6.
n\ = 2d sin d (1516)
The simplest condition to consider will give values for a uniaxial surface
stress or the sum of the principal stresses cri + 02. In the Sachs Weerts^
method two Xray determinations are made of the lattice spacing d for
the Xray beam oriented normal to the specimen surface. One shot gives
the value di for the stressed surface, while the other shot gives do for the
stressfree surface. The lattice constant in the stressfree condition is
obtained either by removing a small plug of metal from the specimen or
by thermally stressrelieving the sample. The strain normal to the
surface which is measured by the X rays is 63.
e, = ^±^ (1517)
do
From the theory of elasticity (Chap. 2), the strain in the direction normal
to the free surface can be expressed as
63 =  i^l + ^ (1518)
Therefore, the sum of the principal stresses lying in the specimen surface
is given by
a, + a,= ^^^ (1519)
A much more general equation for determining the principal stresses
in a surface by X rays can be obtained by considering the general case
for principal stress (Sec. 25). The situation is discussed in detail by
Barrett.^ The generalized state of principal stress acting on the surface
can be represented in three dimensions by an ellipsoid (Fig. 158). The
normal stress given by the coordinates rp and 4> can be expressed in terms
of the three principal stresses and their direction cosines I, m, and n as
1 A. L. Christenson and E. S. Rowland, Trans. ASM, vol. 45, pp. 638676, 1953;
D. P. Koistinen and R. E. Marburger, Trans. ASM, vol. 51, pp. 537555, 1959.
« G. Sachs and J. Weerts, Z. Physik, vol. 64, pp. 344358, 1930.
" C. S. Barrett, "Structure of Metals," 2d ed., chap. 14, McGrawHill Book Com
pany, Inc., New York, 1952.
Sec. 155]
follows,
Residual Stresses
(T = Pai {■ mV2 + nVa
409
(1520)
where I = sin \p cos (f), m = sin i/' sin </>, and n = cos ^. Since the Xray
determination of residual stress considers only the stresses in the surface,
Fig. 158. Representation of principal stresses ai, ai, az by ellipsoid of stress.
03 = 0. Referring to Fig. 158, this requires that yp = 90°. Therefore,
Eq. (1520) can be simplified as follows to give the component of stress
in the direction 4>:
00 = o"! cos^ <^ + 02 sin^ 4> (1521)
An equation exactly analogous to Eq. (1520) can be written for the
principal strains.
e = rei \ m^e.2 + n^e^ (1522)
If the values of the direction cosines, together with the values of d and
€2 in terms of principal stresses are substituted into Eq. (1522), we get
e — 63 = — ^^ — sin x}/ (a I cos^ <^ \ a 2 sin </>)
Substitution of Eq. (1521) into Eq. (1523) leads to the relation
_ e — 63 E
sin''
However,
sin^ xj/ I { V
e — es
d^ — do di — do d^ — di
do
do
(1523)
(1524)
(1525)
where do = atomic spacing in unstressed condition
di = atomic spacing in stressed metal perpendicular to specimen
surface
d^, = atomic spacing in direction defined by angles \p and <f>
410 Applications to Materials Testing [Chap. 1 5
Since the accuracy of Eq. (1525) is chiefly determined by the precision
with which the numerator is known, it is permissible to substitute di
for do in the denominator without greatly affecting the results. This
substitution is an important simplification of the experimental procedure
because it eliminates the necessity for making measurements on stress
free samples. Therefore, the stress in the surface of the specimen for any
orientation of the azimuth angle 4> is related to Xray measurements of
atomic lattice spacing by the following equation:
d^ — di E I
C4. = ^ :f— ] ^^r (1526)
di 1 \ V sm^ \l/
The number of Xray exposures that is required depends on the infor
mation that is available. For any azimuth angle it is customary to
make two exposures to determine the stress in the direction. One
exposure is made normal to the surface {\f/ = 0) to give di, and another
exposure is made for the same value of </>, but with the Xray beam
inclined at an angle \l/ from the normal. Generally Ap — 45° is used for
this second exposure to give the value of d^. If the directions of two
principal stresses oi and a<i in the specimen surface are known, as is often
the case for residual stresses produced by quenching or plastieforming
operations, all that is necessary is to make three exposures to determine
the complete biaxialstress state at the point on the surface. One per
pendicular exposure determines d\, one exposure at i/' = 45° in the direc
tion <^i determines a\ from Eq. (1526), while a third exposure at 1/' = 45°
in the direction of the other principal stress <^2 determines ai from
Eq. (1526). 03 is zero because the measurements are made on a free
surface.
If it is necessary to determine both the magnitude and direction of a\
and C72, it is necessary to make four Xray exposures. Three arbitrary
stress components in three known <^ directions on the surface must be
determined in addition to a normal exposure to determine d\. It is cus
tomary to make these exposures at (^, </> f 60°, and </> — 60°, with s/' = 45°
for each value of 0. The stresses in the three arbitrary directions are
calculated from Eq. (1526) and are then converted into principal stresses
by the methods given in Chap. 2 or by the equations given by Barrett^
for the specific conditions established above.
In addition to the difficulties of accurately measuring the values of
interatomic spacing in highly strained metals which have already been
discussed, there is some uncertainty as to the values of elastic modulus E
and Poisson's ratio v which should be used in Eq. (1526) to calculate
values of residual stress. It is known that for most metals these elastic
1 Ihid., pp. 326327.
Sec. 156] Residual Stresses 411
constants vary considerably with crystallographic direction. Since the
residual stresses calculated from Eq. (1526) are based on Xray measure
ments of lattice strain in certain fixed directions in the crystal lattice,
it is questionable that average values of E and v determined from the
tension test should apply. Experimental data on this point are some
what contradictory. Most investigators who have used Xray methods
for residualstress determination have considered that the use of the
average value of E and v introduce no serious errors beyond that due to
uncontrollable factors. For greatest accuracy, these constants should be
determined for each material and experimental setup by making Xray
measurements on stressrelieved specimens which are subjected to known
\ loads.
1 56. Quenching Stresses
In the introductory discussion of residual stresses in Sec. 151 it was
shown that cooling an ingot of a metal which does not experience a phase
change can produce residual stresses because of the strain mismatch pro
duced by the differential contraction between the cooler and hotter parts
of the body. The development of the longitudinal residualstress pattern
was shown in Fig. 152. Quenching a body from a high temperature
to a lower temperature accentuates the development of residual stresses
because the greater temperature differential which is produced between
the surface and the center due to the rapid rate of cooling produces a
greater mismatch of strain. The situation of greatest practical interest
involves the residual stresses developed during the quenching of steel for
hardening. However, for this case the residualstress pattern is due to
thermal volume changes plus volume changes resulting from the trans
formation of austenite to martensite. The simpler situation, where the
stresses are due only to thermal volume changes, will be considered first.
This is the situation encountered in the quenching of a metal which does
not undergo a phase change on cooling. It is also the situation encoun
tered when steel is quenched from a tempering temperature below the
A I critical temperature.
The distribution of residual stress over the diameter of a quenched bar
in the longitudinal, tangential, and radial directions is shown in Fig. 159a
for the usual case of a metal which contracts on cooling. Figure 159c
shows that the opposite residualstress distribution is obtained if the
metal expands on cooling. The development of the stress pattern shown
in Fig. 159a can be visualized as follows: The relatively cool surface of
the bar tends to contract into a ring that is both shorter and smaller in
diameter than the original diameter. This tends to extrude the hotter,
more plastic center into a cylinder that is longer and thinner than its
412 Applications to Materials Testing
[Chap. 15
original dimensions. If the inner core were free to change shape inde
pendently of the outer ring, it would change dimensions to a shorter and
thinner cylinder on cooling. However, continuity must be maintained
throughout the bar so that the outer ring is drawn in (compressed) in the
longitudinal, tangential, and radial directions at the same time as the
inner core is extended in the same directions. The stress pattern given
in Fig. 159a results.
k^L
Longitudinol
Tangential
I = longitudinal
r = tangential
/? radial
Id)
Radia
Fig. 1 59. Residualstress patterns found in quenched bars, due to thermal strains
(schematic), (a) For metal which contracts on coohng; (6) orientation of directions;
(c) for metal which expands on cooling.
The magnitude of the residual stresses produced by quenching depends
on the stressstrain relationships for the metal and the degree of strain
mismatch produced by the quenching operation. For a given strain mis
match, the higher the modulus elasticity of the metal, the higher the
residual stress. Further, since the residual stress cannot exceed the yield
stress, the higher the yield stress, the higher the possible residual stress.
The yieldstresstemperature curve for the metal is also important. If
the yield stress decreases rapidly with increasing temperature, then the
strain mismatch will be small at high temperature because the metal can
accommodate to thermally produced volume changes by plastic flow.
On the other hand, metals which have a high yield strength at elevated
temperatures, like superalloys, will develop large residual stresses from
quenching.
Sec. 1 56] Residual Stresses 413
The following combination of physical properties will lead to high mis
match strains on quenching:
1. A low thermal conductivity k
2. A high specific heat c
3. A high coefficient of thermal expansion, a
4. A high density p
These factors can be combined into the thermal diffusivity, Dt = k/pc.
Low values of thermal diffusivity lead to high strain mismatch. Other
factors which produce an increase in the temperature difference between
the surface and center of the bar promote high quenching stresses. These
factors are (1) a large diameter of the cylinder, (2) a large temperature
difference between the initial temperature and the temperature of the
quenching bath, and (3) a high severity of quench.
In the quenching of steels, austenite begins to transform to martensite
whenever the local temperature of the bar reaches the Ms temperature.
Since an increase in volume accompanies this transformation, the metal
expands as the martensite reaction proceeds on cooling from the Ms to
Mf temperature. This produces a residual stress distribution of the type
shown in Fig. 159c. The residualstress distribution in a quenched steel
bar is the resultant of the competing processes of thermal contraction
and volume expansion due to martensite formation. Transformation of
austenite to bainite or pearlite also produces a volume expansion, but of
lesser magnitude. The resulting stress pattern depends upon the trans
formation characteristics of the steel, as determined chiefly by its com
position and hardenability, and the heattransfer characteristics of the
system, as determined primarily by the bar diameter, the austenitizing
temperature, and the severity of the quench.
Figure 1510 illustrates some of the possible residualstress patterns
which can be produced by quenching steel bars. The left side of this
figure illustrates a typical isothermal transformation diagram for the
decomposition of austenite. The cooling rates of the outside, midradius,
and center of the bar are indicated on this diagram by the curves marked
0, m, and c. In Fig. 15 10a the quenching rate was rapid enough to
convert the entire bar to martensite. By the time the center of the bar
reached the Ms temperature, the transformation had been essentially
completed at the surface. The surface layers tried to contract against
the expanding central core, and the result is tensile residual stresses at
the surface and compressive stresses at the center of the bar (Fig. 15106).
However, if the bar diameter is rather small and it has been drastically
quenched in brine so that the surface and center transform at about the
same time, the surface will arrive at room temperature with compressive
414 Applications to Materials Testing
[Chap. 15
residual stresses. If the bar is slackquenched so that the outside trans
forms to martensite while the middle and center transform to pearlite
(Fig. 15lOc), there is little restraint offered by the hot, soft core during
the time when martensite is forming on the surface, and the core readily
accommodates to the expansion of the outer layers. The middle and
Austenite
Martensite
log time
(a)
Austenite
Martensite
log time
ic)
+cr
L/^
1
ff^
^
r/
\
/
\
id)
Fig, 1 510. Transformation characteristics of a steel (a and c), and resulting residual
stress distributions {b and d).
center pearlite regions then contract on cooling in the usual manner and
produce a residualstress pattern consisting of compression on the surface
and tension at the center (Fig. 15lOd). Other types of stress distribu
tions are possible, depending upon the cooling rate and the transforma
tion cKaraCteristics of the steel. For example, it is possible to produce a
)ec.
157] Residual Stresses 415
stress pattern with tensile stresses at the surface and center of the bar
and compressive stresses at midradius.
In relatively brittle materials like tool steels, the yield stress is not far
below the fracture stress. Surface tensile stresses equal to the yield stress
may be produced by quenching. If stress raisers are present, the residual
tensile stress may exceed the fracture stress. Cracks produced by tensile
quenching stresses are called quench cracks. In order to minimize quench
cracking in brittle steels, an interrupted quench, or martempering, pro
cedure is sometimes used. The steel is quenched from the austenitizing
temperature into a holding bath which is maintained at a temperature
above the Ms temperature of the steel. The steel is kept at the tem
perature long enough for it to come to thermal equilibrium, and then it is
quenched to a lower temperature to form martensite.
The reasons for tempering hardened steel are to relieve the high micro
stresses in the martensite and to reduce the level of macro residual stress.
Although tempering generally lowers the level of residual stress, there are
exceptions to this behavior.^ For steel bars which have been hardened
through, the level of residual stress will decrease uniformly with increas
ing tempering temperature. For a bar with surface compressive stresses
due to incomplete hardening an increase in tempering temperature will
progressively reduce the residual stress. If a bar is almost completely
hardened through, greater contraction due to tempering of the martensite
will occur at the surface than at the center. This will cause the surface
tensile stress to increase as tempering progresses.
157. Surface Residual Stresses
There are a number of important technological processes which pro
duce high surface residual stresses which have their maximum tvalue
either at the surface or just below the surface and which fall off rapidly
with distance in from the surface. The steep stress gradient produced
by these processes is in contrast with the more uniform residualstress
gradient produced by quenching and most mechanical forming operations.
Normally, these processes produce high surface compressive residual
stresses which aid in preventing fatigue failure. Figure 1511 is typical
of the kind of residualstress distribution which is considered in this
section.
The process of induction hardening consists in inductively heating a
thin surface layer of a steel part above the transformation temperature
and then flashquenching this region to martensite with a spray of water.
The localized expansion of the martensitic case produces a residualstress
pattern of compression stresses at the surface and tension in the interior
1 A. L. Boegehold, Metal Progr., vol. 57, pp. 183188, 1950.
416
Applications to Materials Testing
[Chap. 1 5
Surface
Surface
Fig. 1511.
distribution
Typical residualstress
produced by surface
treatment like induction hardening,
carburizing, or shot peening.
Flame hardening produces the same metallurgical changes and residual
stress pattern as induction hardening, but the local heating is produced
by a gas flame. Both processes not only produce a favorable residual
stress pattern but also leave a hardened surface which improves the
resistance to wear.
Carburizing consists in changing the
carbon content of the surface layers
of a steel part by the diffusion of
carbon atoms into the surface. The
process differs from induction hard
ening in that the entire part is heated
into the austenite region during the
diffusion process and when the part is
quenched volume changes may occur
through most of the cross section. In
general, the carburized surface contains
compressive residual stresses, but con
siderable variation in residualstress
distribution can be produced by changes
in processing. 1 Ordinarily, the maxi
mum compressive residual stress occurs at a depth in from the surface which
is just short of the boundary between the carburized case and the lower
carboncontent core. Nitriding consists in the diffusion of nitrogen into
a steel surface at a temperature below the 4 1 transformation temperature.
The only volume expansion arises from the formation of nitrides at the
surface. The residualstress pattern consists of compression at the sur
face and tension at the interior. Because there is no volume expansion
occurring in the core, as is possible for carburizing, the residualstress
distribution produced by nitriding is easier to control.
Shot peening consists in subjecting a metal surface to the impact of a
stream of fine metallic particles (shot). The impact of the shot on the
surface causes plastic stretching of the surface fibers at a multitude of
localized regions. When this localized plastic flow is relieved, it leaves
the surface in a state of compressive residual stress. In the shot peening
of heattreated steel the maximum compressive residual stress which is
produced is about 60 per cent of the yield strength of the steel. The
maximum residual stress occurs below the surface at a depth of about
0.002 to 0.010 in. Higher values of compressive residual stresses can be
obtained by strain peening. For example, if a flat leaf spring is preloaded
in bending, the resulting compressive residual stress on the surface which
was prestressed in tension will be greater than if the steel was shotpeened
in an initially stressfree condition. The variables of the shotpeening
1 R. L. Mattson, Proc. Intern. Conf. Fatigue of Metals, London, 1956, pp. 593603.
Sec. 158] Residual Stresses 417
process''^ which mfluence the magnitude and distribution of the residual
stresses are (1) the shot size, (2) the shot hardness, (3) the shot velocity,
and (4) the exposure time (coverage).
High surface compressive residual stress can also be developed by sur
face rolling, as was shown in Fig. 151. An estimate of the loads required
for rolling to a given depth may be made by the use of Hertz's theory of
contact stresses.^
Surface rolling is well adapted to the beneficial protection of critical
areas like fillets and screw threads of large parts, while shot peening lends
itself to highproduction volume parts where broad or irregular surfaces
are to be treated. Numerous instances have been reported in the liter
ature where the introduction of beneficial compressive residual stresses
by flame hardening,'' carburizing,^ nitriding,^ shot peening,^ and surface
rolling^ has resulted in improved fatigue performance. Surface working
operations such as shot peening and rolling have also been successfully
used to mitigate the effects of stress corrosion cracking.
1 58. Stress Relief
The removal or reduction in the intensity of residual stress is known as
stress relief. Stress relief may be accomplished either by heating or by
mechanical working operations. Although residual stresses will slowly
disappear at room temperature, the process is very greatly accelerated
by heating to an elevated temperature. The stress relief which comes
from a stressrelief anneal is due to two effects. First, since the residual
stress cannot exceed the yield stress, plastic flow will reduce the residual
stress to the value of the yield stress at the stressrelief temperature.
Only the residual stress in excess of the yield stress at the stressrelief
temperature can be eliminated by immediate plastic flow. Generally,
most of the residual stress will be relieved by timedependent stress relax
ation. Relaxation curves such as those of Fig. 1315 are useful for esti
mating stressrelief treatments. Since this process is extremely temper
1 Ibid.; R. L. Mattson and W. S. Coleman, Trans. SAE, vol. 62, pp. 546556, 1954.
^ J. M. Lessells and R. F. Brodrick, Proc. Intern. Conf. Fatigue of Metals, London,
1956, pp. 617627.
3 J. M. Lessells, "Strength and Resistance of Metals," pp. 256259, John Wiley &
Sons, Inc., New York, 1954.
* O. J. Horger and T. V. Buckwalter, Proc. ASTM, vol. 41, pp. 682695, 1941.
* "Fatigue Durability of Carburized Steel," American Society for Metals, Metals
Park, Ohio, 1957.
6 H. Sutton, Metal Treatment, vol. 2, pp. 8992, 1936.
^ O. J. Horger and H. R. Neifert, Proc. Soc. Exptl. Stress Analysis, vol. 2, no. 1,
pp. 178189, 1944.
8 O. J. Horger, /. Appl. Mech., vol. 2, pp. A128136, 1935.
418 Applications to Materials Testing [Chap. 1 5
aturedependent, the time for nearly complete elimination of stress can be
greatly reduced by increasing the temperature.^ Often a compromise
must be made between the use of a temperature high enough for the
relief of stress in a reasonable length of time and the annealing of the
effects of cold work.
The differential strains that produce high residual stresses also can be
eliminated by plastic deformation at room temperature. For example,
products such as sheet, plate, and extrusions are often stretched several
per cent beyond the yield stress to relieve differential strains by yielding.
In other cases the residualstress distribution which is characteristic of a
particular working operation may be superimposed on the residualstress
pattern initially present in the material. A surface which contains tensile
residual stresses may have the stress distribution converted into beneficial
compressive stresses with a surface working process like rolling or shot
peening. However, it is important in using this method of stress relief
to select surface working conditions which will completely cancel the
initial stress distribution. For example, it is conceivable that, if only
very light surface rolling were used on a surface which initially contained
tensile stresses, only the tensile stresses at the surface would be reduced.
Dangerously high tensile stresses could still exist below the surface.
An interesting concept is the use of thermal stresses to reduce quench^
ing stresses. Since residual stresses result from thermal gradients pro
duced when a part is being quenched, it is possible to introduce residual
stresses of opposite sign by subjecting a cold piece to very rapid heating.
This concept of an "uphill quench" has been used^ in aluminum alloys
to reduce quenching stresses by as much as 80 per cent at temperatures
low enough to prevent softening.
BIBLIOGRAPHY
Baldwin; W. M. : Residual Stresses in Metals, Twentythird Edgar Marburg Lecture,
Proc. ASTM, vol. 49, pp. 539583, 1949.
Heindlhofer, K.: "Evaluation of Residual Stress," McGrawHill Book Company,
Inc., New York, 1948.
Horger, O. J.: Residual Stresses, "Handbook of Experimental Stress Analysis,"
pp. 459469, John Wiley & Sons, Inc., New York, 1950.
Huang, T. C: Bibliography on Residual Stress, SAE Spec. Publ. SP125, 1954;
suppl. 1, SP167, February, 1959.
Residual Stresses, Metal Progr., vol. 68, no. 2A, pp. 8996, Aug. 15, 1955.
"Residual Stress Measurements," American Society for Metals, Metals Park, Ohio,
1952.
Symposium on Internal Stresses in Metals and Alloys, Inst. Metals Mon. Rept. Ser. 5,
1948.
* Typical stressrelief temperatures and times for many metals will be found in
Metal Prog., vol. 68, no. 2A, p. 95, Aug. 15, 1955.
2 H. N. Hill, R. S. Barker, and L. A. Willey, Trans. ASM, vol. 52, pp. 657674, 1960.
Chapter 16
STATISTICS APPLIED TO
MATERIALS TESTING
161. Why Statistics?
There are at least three reasons why a working knowledge of statistics
is needed in mechanical metallurgy. First, mechanical properties, being
structuresensitive properties, frequently exhibit considerable variability,
or scatter. Therefore, statistical techniques are useful, and often even
necessary, for determining the precision of the measurements and for
drawing valid conclusions from the data. The statistical methods which
apply in mechanical metallurgy are in general no different from the tech
niques which are used for analysis of data in other areas of engineering
and science. Mechanical metallurgy is one of the few areas of metal
lurgy where large numbers of data are likely to be encountered, and there
fore it is a logical place in which to introduce the elements of statistical
analysis of data to metallurgists who may have had no other training in
the subject. A second reason for considering statistics in conjunction
with mechanical metallurgy is that statistical methods can assist in
designing experiments to give the maximum amount of information with
the minimum of experimental investigation. Finally, statistical methods
based on probability theory have been developed to explain certain prob
lems in mechanical metallurgy. The explanation of the size effect in
brittle fracture and fatigue is a good example of the use of statistical
theory in mechanical metallurgy.
It is recognized that the material which can be included in a single
chapter can hardly consist of more than an introduction to this subject.
In order to cover the greatest amount of ground, no attempt has been
made to include the mathematical niceties which are usually a part of
a course in statistics. Instead, emphasis has been placed on showing
how statistics can be put to work. Numerous references are included to
sources of more complete discussions and to techniques and applications
419
420 Applications to Materials Testing [Chap. 16
which could not be included because of space limitations. It is hoped
that this chapter contributes to a better appreciation of the usefulness of
statistics in metallurgical research by providing the background neces
sary to see a problem from a statistical viewpoint and enough of the
standard tools of statistics to allow a valid analysis in straightforward
applications. It is important to emphasize that many practical prob
lems are just too complicated to treat by means of the simpler statistical
techniques. In these cases it is important to acquire the services of a
trained statistician for the planning of the experiment and the analysis
of the data.
162. Errors and Samples
The act of making any type of experimental observation involves two
types of errors, systematic errors (which exert a nonrandom bias), and
experimental, or random, errors. Systematic errors arise because of
faulty control of the experiment. Experimental, or random, errors are
due to limitations of the measuring equipment or to inherent variability
in the material being tested. As an example, in the measurement of the
reduction of area of a fractured tensile specimen, a systematic error could
be introduced if an improperly zeroed micrometer were used for measur
ing the diameter, while random errors would result from slight differences
in fitting together the two halves of the tensile specimen and from the
inherent variability of reductionofarea measurements on metals. By
averaging a number of observations the random error will tend to cancel
out. The systematic error, however, will not cancel upon averaging.
One of the major objectives of statistical analysis is to deal quantitatively
with random error.
When a tensile specimen is cut from a steel forging and the reduction
of area is determined for this specimen, the observation represents a sam
ple of the "population from which it was drawn. The population, in this
case, is the collection of all possible tensile specimens which could be cut
from this forging or from all other forgings which are exactly identical.
As more and more tensile specimens are cut from the forging and reduc
tionofarea values measured, the sample estimate of the population
values becomes better and better. However, it is obviously impractical
to sample and test the entire forging. Therefore, one of the main pur
poses of statistical techniques is to determine the best estimate of the
population parameters from a randomly selected sample. The approach
which is taken is to postulate that for each sample the population has
fixed and invariant parameters. However, the corresponding parameters
calculated from samples contain random errors, and therefore the sample
provides only an estimate of the population parameters. It is for this
Sec. 163]
Statistics Applied to Materials Testing 421
reason that statistical methods lead to conclusions having a given proba
hiliiij of being correct.
163. Frequency Distribution
When a large number of observations are made from a random sample,
a method is needed to characterize the data. The most common method
is to arrange the observations into a number of equalvalued class intervals
and determine the frequency of the observations falling within each class
interval. In Table 161, out of a total sample of 449 measurements of
Table 161
Frequency Tabulation of Yield Strength of Steelj
Yield
strength,
1,000 psi,
class
Class
midpoint
Xi
Frequency
/i
fiXi
Frequency,
% of total
Cumulative
frequency
Cumulative
frequency,
%
interval
(1)
(2)
(3)
(4)
(5)
(6)
(7)
114115.9
115
4
460
0.9
4
0.9
116117.9
117
6
702
1.3
10
2.2
118119.9
119
8
952
1.6
18
3.8
120121.9
121
26
3,146
5.8
44
9.6
122123.9
123
29
3,657
6.5
73
16.1
124125.9
125
44
5,500
9.8
117
25.9
126127.9
127
47
5,969
10.5
164
36.4
128129.9
129
59
7,611
13.1
223
49.5
130131.9
131
67
8,777
15.0
290
64.5
132133.9
133
45
5,985
10.0
335
74.5
134135.9
135
49
6,615
10.9
384
85.4
136137.9
137
29
3,973
6.5
413
91.9
138139.9
139
17
2,363
3.8
430
95.7
140141.9
141
9
1,269
2.0
439
97.7
142143.9
143
6
858
1.3
445
99.0
144145.9
145
4
580
0.9
449
99.9
S/i = 449 LfiXi = 58,417
X = 58,417/449 = 130,300 psi
t Data from F. B. Stulen, W. C. Schulte, and H. N. Cummings, in D. E. Harden
bergh (ed.), "Statistical Methods in Materials Research," Pennsylvania State Uni
versity, University Park, Pa., 1956.
yield strength, 4 observations fell between 114,000 and 115,900 psi,
26 fell between 120,000 and 121,900 psi, etc. An estimate of the fre
quency distribution of the observations can be obtained by plotting the
frequency of observations against the class intervals of the yieldstrength
422 Applications to Materials Testing
[Chap. 16
70

60
50

p
1 —
40

30
J
^
20
10

~i
rT
Ti
114 118 122 126 130 134 138 142 146
Yield strength, 1,000 psi
Fig. 161. Frequency histogram of data in Table 161.
70

Mean
Median
xMode
60

/ \
50

/ \
40

/ \
30

/ \
20

/ \
10
 /
\
1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1
114
122 126 130 134 138 142 146
Yield strength, 1,000 psi
Fig. 162. Frequency distribution of data in Table 161.
Sec. 163]
Statistics Applied to Materials Testing 423
measurements (Fig. 161). This type of bar diagram is known as a
histogram. As the number of observations increases, the size of the class
interval can be reduced until we obtain the limiting curve which repre
sents the frequency distribution of the sample (Fig. 162). Note that most
of the values of yield strength fall within the interval from 126,000 to
134,000 psi.
If the frequency of observations in each class interval is expressed as
a percentage of the total number of observations, the area under a fre
quencydistribution curve which is plotted on this basis is equal to unity
(Fig. 163). The probabihty that a single random measurement of yield
strength will be between the value Xi and a slightly higher value Xi + AX
is given by the area under the frequencydistribution curve bounded by
these two limits. Similarly, the probability that a single observation
20

15
^
10
5
 /
1
\
1 1 1 1 1 1
1
1 1 1
1 1 1 1 1 1
114 118 122 126 130 134 138 142 146
Yield strength, 1,000 psi
Fig. 163. Frequency distribution based on relative frequency.
will be greater than some value X2 is given by the area under the curve
to the right of Xo, while the probability that the single observation will be
less than X2 is given by the area to the left of X2.
Another way of presenting these data is to arrange the frequency in a
cumulative manner. In Table 161, column 6, the frequency for each
class interval is accumulated with the values of frequency for the class
intervals below it to indicate the total number of observations with a
value of reduction of area less than or equal to the value of the upper
hmit of the class interval. As an example, for the sample described in
Table 161, 164 out of 449 observations of yield strength have a value of
127,900 psi or less. If the cumulative frequency is expressed as the per
centage of the total (Table 161, column 7), the values represent the
probability that the yield strength will be less than or equal to the value
of the observation. Figure 164 shows the cumulative frequency distri
bution plotted on this basis. The presentation of data as a cumulative
424
Applications to Materials Testing
100
fChap. 16
114 118 122 126 130 134 138 142 146
Yield strength, 1,000 psi
Fig. 164. Cumulative frequency distribution.
distribution is sometimes preferred to a frequency distribution because
it is much less sensitive than the frequency distribution to the choice of
class intervals.
164. Measures of Central Tendency and Dispersion
A frequency distribution such as that of Fig. 162 can be described
with numbers which indicate the central location of the distribution and
how the observations are spread out from the central region (dispersion).
The most common and important measure of the central value of an array
of data is the arithmetic mean, or average. The mean of Xi, X2, • • • ,
Xn observations is denoted by X and is given by
X
Ix.
(161)
The arithmetic mean is equal to the summation of the individual obser
vations divided by the total number of observations, n. If the data are
arranged in a frequency table, as in Table 161, the mean can be most
Sec. 164] Statistics Applied to Materials Testing 425
conveniently determined from
k
X = ^^^ (162)
where /» is the frequency of observations in a particular class interval
with midpoint X,. The summation is taken over all class intervals (see
Table 161, column 4). Equation (162) is an approximation to Eq.
(161).
Two other common measures of central tendency are the mode and the
median. The mode is the value of the observations which occurs most
frequently. The median is the middle value of a group of observations.
For a set of discrete data the median can be obtained by arranging the
observations in numerical sequence and determining which value falls in
the middle. For a frequency distribution the median is the value which
divides the area under the curve into two equal parts. The mean and
the median are frequently close together; but if there are extreme values
in the observations (either high or low), the mean will be more influenced
by these values than the median. As an extreme example, the situation
sometimes arises in fatigue testing that out of a group of specimens tested
at a certain stress a few do not fail in the time allotted for the test and
therefore they presumably have infinite lives. These extreme values
could not be grouped with the failed specimens to calculate a mean
fatigue life; j^et they could be considered in determining the median.
The positions of the mean, median, and mode are indicated in Fig. 162.
The most important measure of the dispersion of a sample is given by
the variance s^.
I (Xi  xy
s' = '^ (163)
w — 1
The term Xi — X is the deviation of each observation Xi from the
arithmetic mean X of the n observations. The quantity n — 1 in the
denominator is called the number of degrees of freedom and is equal to
the number of observations minus the number of linear relations between
the observations. Since the mean represents one such relation, the num
ber of degrees of freedom for the variance about the mean is n — 1.
For computational purposes it is often convenient to calculate the vari
ance from the following relation:
n
n{n — 1)
426 Applications to Materials Testing
[Chap. 16
When the data are arranged in a frequency table, the variance can be
most readily computed from the following equation:
k \ 2
2 fiXi
d/'^')
s' =
n — I
(165)
In dealing with the dispersion of data it is usual practice to work with
the standard deviation s, which is defined as the positive square root
of the variance.
s =
I (X,  xy
1
(166)
Sometimes it is desirable to describe the variability relative to the
average. The coefficient of variation v is used for this purpose.
(167)
A measure of dispersion which is sometimes used because of its extreme
simplicity is the range. The range is simply the difference between the
largest and smallest observation. The range does not provide as precise
estimates as does the standard deviation.
165. The Normal Distribution
Many physical measurements follow the symmetrical, bellshaped
curve of the normal, or Gaussian, frequency distribution. Repeated
measurements of the length or diameter of a bar would closely approxi
mate this frequency distribution. The distributions of yield strength,
tensile strength, and reduction of area from the tension test have been
found to follow the normal curve to a suitable degree of approximation.
The equation of the normal curve is
viX)
1
^ exp
aVSTT
\{^)
(168)
where p(X) is the height of the frequency curve corresponding to an
assigned value X, fx is the mean of the population, and <y is the standard
deviation of the population.^
The normal frequency distribution described by Eq. (168) extends
from X = — octoX= +oo and is symmetrical about fx. The constant
It should be noted that o is used for normal stress in the other chapters of this book.
Sec. 165]
Statistics Applied to Materials Testing 427
l/o \/27r is used to make the area under the curve equal unity. Figure
165 shows the standardized normal curve when /x = and o = 1. The
parameter z, defined by Eq. (169), is called the standard normal deviate.
z = ^^^^ (169)
For this case, the equation of the normal curve becomes
(1610)
The total area under the curve in Fig. 165 is equal to unity. The rela
tive frequency of a value of z falling between — oo and a specified value
pi
/)
OA
%
/ 0.3
A
0.2
 \
A^/////
0.1
 \
...mB.
1 1 1— _
3
1
Fig. 165. Standardized normal frequency distribution.
is given by the area under the curve between these limits. Table 162
lists some typical values.^ As an example, the area shown shaded in
Fig. 165 is 0.16 the total area under the curve. Further, if ju = 20 and
(7 = 5, what proportion of the sample observations would fall between
X = 20 and X = 10? The respective values of z are and —2, so that
the area under the normal curve between these limits is
0.50  0.02 = 0.48
Reference to Table 162 will also show that approximately 68 per cent
of the area lies in the limits /x ± o", 95 per cent of the area is within the
limits /x + 2o, while 99.7 per cent of the area is covered by the limits
/X + 3(7.
^ For complete tables see W. J. Dixon and F. J. Massey, Jr., "Introduction to Sta
tistical Analysis," 1st ed., p. 306, McGrawHill Book Company, Inc., New York, 1951.
428 Applications to Materials Testing [Chap. 16
Table 162
Areas under Standardized Normal Frequency Curve
z = X — n/a Area
3.0
0.0013
2.0
0.0228
1.0
0.1587
0.5
0.3085
0.0
0.5000
+0.5
0.6915
+ 1.0
0.8413
+2.0
0.9772
+3.0
0.9987
A quick method for determining whether a sample frequency distri
bution is a normal distribution is to plot the observed cumulative fre
quency distribution on normalprobability paper. ^ If the data plot is
an approximate straight line, then the distribution can be considered to
be a normal frequency distribution. Figure 166 shows the data from
Table 161 plotted on probability paper. The mean and standard devi
ation can be readily determined from this type of plot. The mean is the
value of the abscissa corresponding to a cumulative frequency of 50 per
cent. Since 68 per cent of the area under the standardized normal curve
is included within plus or minus one standard deviation of the mean,
it follows that the standard deviation can be obtained from Fig. 166
by taking half the intercept of the abscissa between 84 and 16 per cent.
This method of determining the mean and standard deviation is often
accurate enough for engineering purposes. However, it can be used only
when the data plot as a fairly good line on probability paper. The sta
tistical advantages of the normal distribution are quite considerable, so
that in cases where the data do not fit the normal curve it is often worth
while to search for a simple transformation which will normalize the data.
The most common transformations are X = log X and X = X''^. As
an example, the distribution of fatigue life at constant stress is skewed
with respect to the variable of number of cycles to failure, but it is
approximately normal with respect to the logarithm of the number of
cycles to failure. This is called a lognormal distribution.
The mean of a sample provides an unbiased estimate of the mean of
the population, or universe, from which it was drawn. If repeated sam
^ An analytical method for determining how well a sample frequency distribution
compares with the normal distribution is the chisquare test for goodness of fit. This
technique is discussed in any standard statistics text. For example, see ibid., pp.
190191.
2 A complete description of the lognormal distribution is given by A. Hald, "Sta
tistical Theory with Engineering Applications," pp. 160174, John Wiley & Sons,
Inc., New York, 1952.
165]
Statistics Applied to Materials Testins
429
pies are taken from the population, they will each in general yield a
different value of mean A" and standard deviation s. The sample values
of the mean will, however, be normally distributed about the population
mean fx. The centrallimit theorem of statistics provides a method for
estimating the population mean n and standard deviation a. According
to this theorem, if A" is the mean value of a random sample of n observa
tions A is normally distributed about the population mean with a stand
ard deviation (r/\/n. With this as a basis, it is possible to determine the
confidence limits within w^hich the sample mean approaches the population
mean to a certain level of certainty.
99.99
99.9
99
. 95
I 90
I 80
■^ 70
I 60
50
= 40
1 30
20
o
■t 10
I 5
I 2
1
0.5
0.2
0.1
0.05
0.01

/
\^
/
/
/
/
<<
/
^
/
J
/
/
9'
A>
y
^
114
122 126 130 134 138
Yield strength, 1,000 psi
142 146
Fig. 166. Normalprobability plot of data in Table 161.
For samples with many observations (w > 30) or where the population
variance is known (which is rarely the case), the determination of the
confidence limits of the mean is based on the standard normal deviate z.
For example, suppose that we want to know the interval about a meas
ured sample mean which will include the population mean 95 per cent
of the time, i.e., the 95 per cent confidence level. Reference to an
enlarged version of Table 162 would show that 95 per cent of the area
under the normal curve would be included within the interval [i ± 1.96ct.
Now, since the centrallimit theorem states that the observations of A
430 Applications to Materials Testing
[Chap. 16
are normally distributed about fx with a standard deviation (x/\/n, we can
conclude that 95 per cent of the time the mean of the population will be
within the interval X ± 1.96a/\/n. Since the sample estimate of the
variance s cannot be substituted for the population variance o^ when the
sample size is small, it is necessary to base the calculation^ on the sam
pling distribution of the statistic known as t. From the standpoint of
calculation, this means only that the constant changes with sample size.
For a given confidence level the true mean of the population lies within
the interval X ± ts/\/n, where values of t are obtained from Table 163.
Table 163
Values of t for Determining the Confidence Limits of X and s
Confidence
Sample size n
level, %
3
5
7
10
12
15
20
00
90
95
99
2.35
3.18
5.84
2.01
2.57
4.03
1.89
2.36
3.50
1.81
2.23
3.17
1.78
2.18
3.05
1.75
2.13
2.95
1.72
2.09
2.85
1.64
1.96
2.58
The confidence limits for the standard deviation are best obtained from
a table of the chisquare (x^) distribution. The limits within which the
population variance a^ will lie, based on the sample variance s^, are given
by
<a' <
X?
(1611:
Values of x^ are taken from tables of the distribution for the degrees of
freedom equal to n — 1. For example, if the limits within which s
approaches a^ are required to a 95 per cent level of confidence, then xi" is
taken from the table at a probability of 0.975, while X2^ is taken at the
0.025 probability level. An approximation which is good for large sam
ple sizes is that the true standard deviation lies within the interval s +
ts/\/2n, where the same values of t listed in Table 163 apply.
166. Extremevalue Distributions
The argument is sometimes raised that the normal frequency distri
bution is unsuited for describing mechanical property data because the
long tails on each side of the mean are unrealistic and do not represent
the observed facts. However, for many practical problems involving
statistics these tails can be ignored without affecting the results. More
over, it is possible to set arbitrary lower and/or upper limits to the normal
^ For a discussion of this point see Dixon and Massey, op. at., pp. 97101.
Sec. 166] Statistics Applied to Materials Testins 431
distribution' without destroying its statistical usefulness. The central
limit theorem shows that a normal frecjuency distribution is to be expected
when the effect being observed results from averaging the effects of a
whole series of variables. However, if the effect being observed is due
to the smallest or largest of a number of variables, an extremevalue distri
bution is to be expected. If failure is due to one weakest link or to a
number of more or less equally weak links, the extremevalue distribution
would apply.
The application of the extremevalue distribution to engineering prob
lems has been largely the result of the work of E. J. Gumbel.^ This
theory has been used to predict the maximum wind velocity or the crest
of floods (largest values) and the fatigue life for a 0.1 per cent probability
of failure (smallest value). While the mathematical details of the
extremevalue theory are outside the scope of this chapter, an idea of
the concept can be obtained from an example. Consider a group of
fatigue specimens tested to failure at a constant stress. The distribution
of fatigue life obtained from this sample can be considered to be an
extremevalue distribution if it is assumed that each specimen which
failed represents the weakest out of a large number of specimens tested
to the same particular number of cycles of stress reversal. Methods of
using the extremevalue distribution with fatigue data have been devel
oped,^ and extremevalue probability paper is available for plotting the
results.
A special type of extremevalue distribution is the Weibull distri
bution.^ For this frequency distribution, the cumulativedistribution
function is given by
(^■)
P(X) = 1  exp  ^^^^ (1612)
where P(X) is the probability that a value less than X will be obtained,
Xn is the value of X for which P(X) = 0, and Xo and m are empirically
determined constants. This equation gives a cumulativedistribution
curve which resembles that for a normal distribution except that Weibull's
distribution has a finite lower limit X„. Weibull's distribution has been
used to describe the yieldstrength distribution of steel and the fatigue
1 For a discussion of the truncated normal distribution, see Hald, op. cit., pp. 144
151; A. C. Cohen, J. Am. Statist. Assoc, vol. 44, pp. 518525, 1949.
2 E. J. Gumbel, Statistical Theory of Extreme Values and Some Practical Applica
tions, Natl. Bur. Standards Appl. Math. Ser. 33, Feb. 12, 1954; Probability Tables for
the Analysis of Extremevalue Data, Natl. Bur. Standards Appl. Math. Ser. 22, July
6, 1953; see also J. Lieblein, NACA Tech. Note 3053, January, 1954.
3 A. M. Freudenthal and E. J. Gumbel, Proc. Intern. Conf. Fatigue of Metals, Lon
don, 1956, pp. 262271.
' W. Weibull, J. Appl. Meek., vol. 18, pp. 293297, 1951, vol. 19, pp. 109113, 1952.
432 Applications to Materials Testing [Chap. 16
life at constant stress. It appears to be particularly well suited to
describing the distribution of fatigue life of bearings. ^
When a mass of mechanical test data is analyzed to select reliable
design values, the design engineer is usually interested in establishing
the smallest values of the property being tested, so that the design can be
based on conservative yet realistic strength values. For example, the
designer might be interested in knowing the value of yield strength which
would be exceeded by 99.9 per cent of the specimens, while the metal
lurgist who is trying to improve the yield strength by heat treatment
would be mainly interested in knowing the mean and standard deviation
of his test values. Of course, the value of yield strength which would be
exceeded by 99.9 per cent of the specimens can be calculated from the
sample observations provided that the frequency distribution of the obser
vations is known and the observations faithfully follow this distribution.
Unfortunately, although most mechanical property data fit a normal or
lognormal frequency distribution over the central region of probability,
at the tails or extremes of the distribution appreciable deviations usually
occur. A mean and standard deviation can be determined from any set
of observations by the methods described previously, regardless of the
nature of the frequency distribution, and even if the observations deviate
considerably from a normal distribution, the powerful statistical tests
described in the next section can still be used with relatively little error.
On the other hand, estimates of the probability of failure are not valid
outside the regions in which the data are known to fit the frequency
distribution. The prediction of values at low probabilities of failure,
e.g., the 0.1 per cent level, on the basis of the assumption that the data
follow the normal distribution in this region can lead to considerable
error. Further, because of the large number of tests required to make
an experimental determination of the stress which gives a 1 or 0.1 per
cent probability of failure, there are few experimental observations in
the extreme regions which can be used for guidance. While extreme
value statistics have not yet met with universal acceptance, this approach
is the most reasonable in deahng with small probabilities of failure.
167. Tests of Significance
An important feature of statistical analysis is the ability, through tests
of significance, to make comparisons and draw conclusions with a greater
assurance of being correct than if the conclusions had been based only on
intuitive interpretation of the data. Most of the common statistical tests
of significance are based on the use of the normal distribution, so that it
^ J. Lieblein and M. Zelen, J. Research Natl. Bur. Standards, vol. 57, pp. 273316,
1956.
Sec. 167] Statistics Applied to Materials Testing 433
is a distinct advantage if the test data conform to this frequency distri
bution. A common problem which is readily treated by means of tests
of significance is to determine whether the means and/or standard devi
ations of two samples drawn from essentially the same population really
differ significantly or whether the observed differences are due to random
chance. In statistics the term significant difference is used much more
precisely than in everyday usage. In determining whether or not two
statistics are significantly different it is always necessary to specify the
probability of the test predicting an erroneous conclusion. This is known
as the level of significance. For example, if the test of significance is
made at the 5 per cent level of significance there is 1 chance in 20 that
an erroneous conclusion would be reached by the test.
Difference between Two Sampie Standard Deviations
It is often of interest to know whether or not the variability of one set
of measurements differs significantly from that of another set of data.
For example, it may be important to know whether or not a certain
metallurgical condition shows more scatter in fatigue life than another
treatment or whether or not the scatter is significantly different at high
stresses and at low stresses. The test of significance which is used for
this situation is the F test.
Let Si be the standard deviation for a sample size Wi from the first
population, and let S2 and n^ be the same terms for the second population.
The following ratio is determined, where Si^ > §2^:
F = 'A (1613)
The F ratio obtained from Eci. (161.3) is compared with tables of the
F distribution.' The table corresponding to the selected level of signifi
cance is used. This table is based on the number of degrees of freedom,
which for this case is one less than the sample size. The table is entered
across the top for the degrees of freedom for the numerator in the F ratio
and along the lefthand edge for the degrees of freedom for the denomi
nator. Where this column and row intersect, we get the value of the
F ratio. If the ratio calculated in Eq. (1613) is greater than the F ratio
obtained from the table, the two sets of standard deviations are signifi
cantly different at the chosen level of significance. Note that, as the
level of significance is raised (less risk of making an erroneous decision),
the greater must be the F ratio determined from Eq. (1613). For exam
ple, for ni = no = 21, the F tables give 2.12 at the 5 per cent level and
2.94 at the 1 per cent level of significance.
1 M. Merrington and C. M. Thompson, Biometrika, vol. 33, p. 73, 1943; see also
Dixon and Massey, op. cit., pp. 310313.
434 Applications to Materials Testing [Chap. 16
Difference between Two Means
Sample Standard Deviations Not Significantly Different. If the F test
shows that the standard deviations of the two samples are not signifi
cantly different, the following procedure can be used to determine whether
or not there is a significant difference between the means of the two sam
ples Xi and X2 : First, the variance of the two samples in common should
be determined.
2 ^ (ni — l)gi^ h (n2  l)s2^
^1 + W2 — 2
Then a value for the t statistic can be determined.
(1614)
If the value of t determined from Eq. (1615) is greater than the value
obtained from tables of the t distribution^ at the specified level of signifi
cance and with the degrees of freedom equal to ni \ n^ — 2, there is a
significant difference between the means.
Sample Standard Deviations Significantly Different. If the F test shows
that the standard deviations of the two samples are significantly different,
it is not correct to determine a common variance. The value of t is
determined from Eq. (1616), and a value of c is calculated from Eq.
(1617) to use in determining the number of degrees of freedom (d.f.)
which is used with the t tables.
^ = / 2/ ^ ^ — 27~ul (1616)
c = ./^Y''\/ (1617)
d.f. =
siV^i + S2^/n2
^ (1  cr
(1618)
ni — 1 n2 — 1
N onparametric Tests of Significance
The F test and the t test depend on the requirement that the popu
lations are a good approximation to the normal distribution. Techniques
which do not depend on the mathematical form of the frequency distri
bution are available. These nonparametric methods may be useful in
cases where the data are too meager to allow an estimate of the frequency
distribution to be made. However, because they do not depend on any
sampling distribution, they are less powerful than the methods which
have been described above. Only a brief description of some of the non
1 Ibid., p. 307.
Sec. 168] Statistics Applied to Materials Testins 435
parametric techniques can be given here. For details, reference should
be made to the original sources.
The rank test^ arranges the two samples in numerical order and assigns
rank numbers. The totals of the rank for each sample are compared
with tables to determine whether or not a significant difference exists.
The run test" arranges all values from both samples in ascending order.
The number of "runs" from each sample are determined, and by com
paring with tables, it can be found whether or not the observations of
both samples have been drawn from the same population. To illus
trate the meaning of the term run, if two samples A and B are
arranged in order to give the series ABA ABB A A ABB, there are six
runs ABAABBAAABB in this series.
The confidence interval for the median of a population can be obtained
very readily without any consideration of the distribution function of the
population. If the observations are listed in ascending order, we know
with great certainty that the median of the population lies between the
smallest and largest observation. As we approach the central value of
the sample, our degree of certainty decreases. Tables are available^ for
predicting at various levels of confidence and for different sample sizes
what values in the array the population median would be expected to lie
between. For example, if n = 15, the fourth and twelfth observations
are confidence limits of more than 95 per cent for the population median,
while if n = 50, the population median will lie between the eighteenth
and thirtythird observation.
168. Analysis of Variance
The statistical tests of significance discussed in the previous section
deal only with the analysis of a single factor at two different levels. For
example, by using the methods of the previous section, one could deter
mine whether the transverse reduction of area is significantly different
for different forging reductions or for different heat treatments. How
ever, to determine the interaction between the two requires the analysis
of variance. One method* of analysis of variance will be described below
by means of a simplified illustrative example.
Table 164 gives the values of RAT for a hypothetical situation where
tensile tests were determined from forgings with 10:1 and 30:1 forging
' F. Wilcoxon, "Some Rapid Approximate Statistical Procedures," American
Cyanamid Co., New York, 1949.
2 Dixon and Massey, op. cit., pp. 251255.
3 Ibid., table 25, p. 360.
* Based on a procedure given by C. R. Smith, Metal Progr., vol. 69, pp. 8186,
February, 1956.
436
Applications to Materials Testing
[Chap. 16
reductions, with and without a homogenization heat treatment. We are
interested in answering the following questions.
1. Is there a significant difference in RAT for a 10: 1 and 30: 1 forging
reduction?
2. Is there a significant difference in RAT between a forging which was
given no homogenization treatment and one given the homogenization
treatment?
3. Is there a significant interaction between forging reduction and
homogenization which affects the level of RAT?
Table 164
Effect of Forging Reduction and Homogenization Treatment on
Transverse Reduction of Area (RATj
Homogenization
Forging reduction
10:1
30:1
None
25.8
30.4
28.6
35.6
20.3
22.8
19.5
30.6
28.5
25.5
Ti
y. 140
7
Ti
,. 126.9
2400°F, 50 hr
26.4
35.6
30.2
21.4
27.2
30.6
35.3
27.9
28.2
31.5
Ti
i. 140
8
Ti
y. 153.5
T.i. 267.6
281.5
280.4
T.j. 294.3
561.9
For the arrangement of the data shown in Table 164 vertical listings
are called columns and horizontal listings are called rows. The experi
mental values for each condition constitute a cell. The total of all the
values in each cell is indicated by {Ta). The total for each forging
reduction irrespective of homogenization is called {Ti.). The total for
each homogenization condition regardless of forging reduction is (T.j.).
(T...) is the total of all values in the table. Further definitions are as
follows :
k = number of columns in table = 2
I = number of rows in table = 2
m = number of readings in each cell = 5
Sec. 168] Statistics Applied to Materials Testing 437
The following computations are made from Table 164:
1. Total sum of squares (TSS)
where ^X^j is the summation of the square of all readings, or
TSS = 16,208.67  J^l;^l\ = 422.09
2. Subtotal sum of squares (SSS)
4^  7^ (1620)
m klm.
ggg ^ (140.7)^ + (126.9) + (140.8)^ + (153.5)^ _ (561.9)'^ ^ ^^ ^^^
5 20 ■ "
3. Withincells sum of squares (WSS)
WSS = TSS  SSS (1621)
WSS = 422.09  70.82 = 351.27
4. Betweencolumns sum of squares (BCSS)
y 7^2 712
BCSS = ^  if (1622)
Im klm
^^^^ (281.5)^ + (280.4)^ (561.9)^ „_
5. Betweenrows sum of squares (BRSS)
y /T12 7^2
BRSS = ^  fy (1623)
km klm
(267.6)^^ + (294.3)=^ (561.9)^ _
^^'^^ = 2IO 20^ "
6. Interaction sum of squares (ISS)
ISS = SSS  BCSS  BRSS (1624)
ISS = 70.82  0.06  35.65 = 35.11
Once the computations are made, the results can be entered in an
analysisof variance table such as Table 165. Values of mean square
are obtained by dividing the values of sum of the squares by their respec
tive degrees of freedom. To determine whether or not there are signifi
438
Applications to Materials Testing
[Chap. 16
cant differences between the columns and rows and interactions between
the two, the respective mean squares are used in the F test. For example,
to determine whether or not there is a significant interaction between
forging reduction and homogenization treatment, the following F ratio is
determined :
interaction mean square
F =
F =
withincells mean square
35.11
21.92
= 1.61
A check of tables of the F distribution at the 5 per cent level of signifi
cance shows that the F ratio must exceed 4.49 for the variability to be
significant. Therefore, there is no significant interaction between these
two variables.
Table 165
Analysis of Variance
Source of variability
Degrees of
freedom
Sum of
squares
Mean
square
Between columns (forging reduction)
Between rows (homogenization)
Interaction between columns and rows
Within cells
1
1
1
16
0.06
35.65
35.11
351.276
0.06
35.65
35.11
21.92
Total
19
To determine whether or not there is a significant effect of homogeni
zation on the level of RAT, the following F ratio is determined:
F =
F =
betweenrows mean square
interaction mean square
35.65
35.11
1.016 i^(0.05) = 161.00
Since the calculated F ratio is far less than the value of F at the 5 per cent
level of significance, it must be concluded that the data show no signifi
cant effect of homogenization on RAT. In the same way it can be shown
that there is no significant difference between the two forging reductions.
169. Statistical Design of Experiments
The greatest benefit can be gained from statistical analysis when the
experiments are planned in advance so that data are taken in a way
which will provide the most unbiased and precise results commensurate
Sec. 169] Statistics Applied to Materials Testing 439
with the desired expenditure of time and money. This can best be done
through the combined efforts of a statistician and the engineer during
the planning stage of the research project. Only a brief introduction to
commonly used statistical designs of experiments can be given in this
section. For details the reader is referred to several excellent texts on
this subject.^
In any experimental program involving a large number of tests, it is
important to randomize the order in which the specimens are selected for
testing. By randomization we permit any one of the many specimens
involved in the experiment to have an equal chance of being selected for
a given test. In this way, bias due to uncontrolled secondorder varia
bles is minimized. For example, in any extended testing program errors
can arise over a period of time owing to subtle changes in the character
istics of the testing equipment or in the proficiency of the operator of the
test. In taking metal specimens from large forgings or ingots the possi
bility of the variation of properties with position in the forging must be
considered. If the objective of the test is to measure the average proper
ties of the entire forging, randomization of the test specimens will mini
mize variability due to position in the forging.
One way of randomizing a batch of specimens is to assign a number to
each specimen, put a set of numbered tags corresponding to the specimen
numbers in a jar, mix them thoroughly, and then withdraw numbers from
the jar. Each tag should be placed back in the jar after it is withdrawn
to allow an equal probability of selecting that number. The considerable
labor which is involved in this procedure can be minimized by using a
table of random numbers. The first number is selected by placing your
finger on a number with your eyes closed. For two digit numbers,
the next number in the table will be found on the page corresponding
to the first digit of the number in the column given by the second digit.
This procedure is repeated over and over until all the specimens are
selected.
In the early stages of a research project it is frequently important to
determine what are the important factors in the problem under study
and what is the relative importance of each of these factors. The use of
a factorial design in place of the conventional approach of changing each
factor one at a time will provide a more efficient answer. Assume that
the problem is to determine the effect of boron and nickel content on the
yield strength of a highstrength steel. In the first trial each factor will
be investigated at two levels. Table 166 indicates that three runs would
' K. A. Brownlee, "Industrial Experimentation," 4th ed., Chemical Publishing
Company, Inc., New York, 1952; O. L. Davies, "The Design and Analysis of Indus
trial Experiments," Oliver & Boyd, Ltd., Edinburgh and London, 1954; W. G. Cochran
and G. M. Cox, "Experimental Designs," John Wiley & Sons, Inc., New York, 1950.
440
Applications to Materials Testing
[Chap. 16
be required by the conventional approach of varying each factor one at
a time. By comparing run 1 with run 2, we can find the effect of varying
the boron content at constant nickel content. Comparing run 1 with
run 3 gives the effect of varying the nickel content for a constant boron
content. For this design, one control experiment, run 1, is used to judge
Table 166
Design of Experiment Based on Conventional Oneatatime Variation
OF Each Factor
Factor
B, %
Ni, %
Run 1
Run 2
Run 3
0.002
0.020
0.002
1.0
1.0
3.0
the effect produced by varying the levels of boron and nickel content.
No information is provided on the effect of boron and nickel when they
are both at their upper level. Furthermore, it is quite possible that the
effect of changing the boron content from the low to the high level would
not be the same at both the low and the high level of nickel content.
In other words, there may be an interaction between the effect of boron
and nickel. A simple design of experiment such as is given in Table 166
cannot determine the existence of interaction.
The factorial design of the experiment for two variables at two levels
is given in Table 167. This design has several advantages over that
Table 167
Factorial Design of the Experiment
Factor
B, %
Ni, %
Run 1
0.002
1.0
Run 2
0.002
3.0
Run 3
0.020
1.0
Run 4
0.020
3.0
given in Table 166. First, more precise statements can be made about
the effect of varying boron and nickel content. All four runs are used in
measuring this main effect by comparing the two runs made at the low
level with the two runs made at the upper level. Second, the interaction
V^etween boron and nickel is determined from a comparison of the differ
ence in results of runs 1 and 4 with the difference in results of runs 2
and 3. An analysis of variance is used to make these comparisons.
Sec. 16 10] Statistics Applied to Materials Testing 441
The illustration given above represents the simplest case of a factorial
design. The number of runs required in a factorial design is equal to
the number of levels raised to the power of the number of factors being
investigated. Thus, if we were interested in four variables at four levels,
256 runs would be required. The very rapid rate at which the number
of runs required by a factorial design increases with small increases in
the number of either variables or levels is the chief disadvantage of this
type of experimental design. However, the disadvantages of simple fac
torial design are overcome to a great extent by a modification known
as fractional replication.^ The number of runs required is substantially
reduced without greatly affecting the power of the method. The main
sacrifice is in runs which, if included, would provide information on
higherorder interactions between the variables. The construction of
fractional replicate designs is covered in the foregoing reference and in a
number of others.^
1610. Linear Regression
The term regression is used in statistics to refer to the determination
of a functional relationship between one or more independent variables
Xi and a dependent variable Yi. Regression techniques are ordinarily
used where it is assumed that a fixed but unknown relationship exists
between the X and Y populations but the random errors in the measure
ments prevent a reliable determination of the relationship by inspection.
An example might be the determination of the relationship between
tensile strength (F) and Brinell hardness (.Y). In this section only the
simplest case of linear regression will be considered. This is also called
the method of least squares because the line which results from the analy
sis has the property that the sum of the squares of vertical deviations of
observations from this line is smaller than the corresponding sum of the
squares of deviations from any other line. Multiple regression, where
there are two or more independent variables in the regression equation,
is an extension of this analysis, and curvilinear regression, where a curved
line is fitted to the data, is a further refinement.
Frequently, the term correlation is used in conjunction with regression.
A clear distinction should be made between regression and correlation.
Regression deals with the situation where there is a clear distinction
between the dependent and independent variables, while correlation deals
with the relationship between two or more sets of data which vary jointlj'.
' D. J. Finney, Ann. Eugenics, vol. 12, pp. 291301, 1945.
2 K. A. Brownlee, B. K. Kelly, and P. K. Loraine, Biometrika, vol. 35, pts. Ill, IV,
pp. 268276, 1948; A. L. Davies and W. A. Hay, Biometrics, vol. 6, no. 3, pp. 233249,
1950.
442 Applications to Materials Testing [Chap. 16
A linearregression equation between the dependent variable Y and
the independent variable X is given by
Y = a + b(X  X) (1625)
i = n
where a = '^ — = Y (1626)
n
n I X.Y,  (I X,)\l F.)
b = ^^ '^^ i=i (1627)
nlX^(l X.y
1=1 1=1
The measure of the variation of the observed value of Y from the value
calculated from the regression equation is called the standard error of
estimate Syx. It plays the role of the standard deviation in regression
analysis.
s,. = ' (s,2  b's.') (1628)
The quantity which determines how well the regression equation fits the
experimental data is the correlation coefficient r. A value of r close to zero
indicates that the regression line is incapable of predicting values of Y,
while a value close to unity indicates nearly perfect prediction.
n n n
n I X.F,  J X, 2 Y,
r = ^^ ^^^ '^ — (1629)
ft Ox<>y
1611. Control Charts
A statistical technique which has proved very useful for routine analy
sis of data is the control chart. The use of this technique is based on the
viewpoint that every manufacturing process is subject to two sources of
variation, chance variation and variation due to assignable causes. The
control chart is a graphical method of detecting the presence of variation
which is greater than that expected as a result of chance. The control
chart is one important method which is used in the branch of applied
statistics known as quality control.^
The use of the control chart can best be described by means of an
1 E. L. Grant, "Statistical Quality Control," 2d ed., McGrawHill Book Company,
Inc., New York, 1952.
Sec. 1611]
Statistics Applied to Materials Testing 443
illustration. Consider a commercial heattreating operation where bear
ing races are being quenched and tempered in a conveyortype furnace
on a continuous 24hr basis. Every 2 hr the Rockwell hardness is meas
ured on 10 bearing races to determine whether or not the product con
forms to the specifications. The mean of each sample X is computed
and the dispersion is determined by computing the range R. A separate
control chart is kept for the mean and the range (Fig. 167). The aver
age of the sample means, /x, and the mean of the sample values of the
47
46 
45
43
42
41
40
Rockwell C hardness
UCL
LCL
J I I L
2 4 6 8 10 12 14
Batch number
UCL
LCL
2 4 6 8 10 12 14
Batch number
Fig. 167. Control charts for X and R.
range, R, are first determined. Next, the control limits are determined.
Points which fall outside the control limits indicate that the variation is
greater than would be expected solely from chance and that corrective
action should be taken to bring the process into control.
The determination of the upper control limit (UCL) and the lower
control limit (LCL) for the mean and the range is greatly simplified by
the use of tables developed by Shewart. ' The control limits for the range
and the means for a 1 per cent chance of the sample value exceeding the
control limits due to random chance can be determined from Table 168.
' W. A. Shewart, "Economic Control of Quality of Manufactured Product," D. Van
Nostrand Company, Inc., Princeton, N.J., 1931.
444 Applications to Materials Testing
For the range
For the means
UCL = DuR
LCL = DlR
UCL = M + A^R
LCL = M  A2R
[Chap. 16
(1630)
(1631)
Table 168
Multipliers for Use in Determining Control Limits
OF Control Charts f
Sample
size
Dl
Do
A2
2
0.009
3.52
1.88
4
0.185
2.26
0.73
6
0.308
1.97
0.48
8
0.386
1.84
0.37
10
0.441
1.76
0.31
12
0.482
1.70
0.27
15
0.524
1.64
0.22
t W. J. Dixon and F. J. Massey, Jr., "Introduction to Statistical Analysis," 1st ed.,
p. 113, McGrawHill Book Company, Inc., New York, 1951.
1612. Statistical Aspects of Size Effects in Brittle Fracture
Mechanicalstrength measurements of brittle materials, like glass and
ceramics, and of metals, under conditions where they behave in a brittle
manner, show a dependence of strength on the size of the test specimen.
As the size of the specimen decreases, the fracture stress in tension, bend
ing, torsion, or impact increases. This is known as a size effect. The
existence of a size effect in fatigue has already been considered in Chap. 12.
An explanation for the size effect can be provided by statistical reason
ing. If it is assumed that brittle fracture is controlled by a distribution
of imperfections or cracks, the fracture stress is given by the Griffith
criterion.
/2Ey
\ ire
r
(1632)
For every size of crack c there will be a certain fracture stress ac given
by Eq. (1632). It is assumed that the specimen is made up of many
volume elements which each contains a single crack. Since it is assumed
that there is no interaction between the cracks in the different volume
elements, the strength of the specimen is determined by the element with
the longest crack, for this results in the lowest value of fracture stress.
Sec. 1612] Statistics Applied to Materials Testing 445
Therefore, the brittlefracture strength is determined, not by an average
value of the distribution of imperfections, but by the one, most dangerous
imperfection. This concept of brittle fracture is called the weakestlink
concept in direct analogy with the fact that the strength of a chain is
determined by the strength of its weakest link. The existence of a size
effect arises quite naturally from this concept. If the crack density of
the material is assumed constant, as the volume of the specimen increases
the total number of cracks also increases and therefore the probability of
encountering a severe crack is increased. If the stress distribution
imposed by the test method is nonuniform, as in bending or torsion, the
analysis must be based on the surface area.
The problem of relating the crack distribution to the fracture strength
is one of finding the distribution function of the smallest value of strength
as a function of the number of cracks A'' for a given distribution function
of crack sizes. Frequently, the distribution of crack size is assumed to
be Gaussian. This implies that the probability of having a small defect
is the same as the probability of occurrence of a large defect. The
mathematics for this crack distribution has been worked out by Frenkel
and Kontorova.^ For a Gaussian distribution of the strength of the
weakest element the strength should decrease linearly with increasing
(log Vy^, where V is the volume of the specimen. WeibuU's distribution
function (see Sec. 166) has also been used to describe the distribution
of crack sizes. ^ This predicts that the strength should decrease with
increasing volume according to V~'^''", where m is the experimentally
determined factor in WeibuU's distribution function. Experiments on
the size effect in tension and bending of steel at low temperature show
good agreement with WeibuU's prediction.^ It is also argued that a
more reasonable distribution of defects decreases in proportion with the
size of the defect. The Laplacian distribution adequately expresses this
requirement.
p{c) =expf^^ (1633)
where p(c) = probability of occurrence of a crack of length c
a = a constant
The Laplacian distribution predicts that the strength decreases linearly
with an increase in log V. Figure 168 shows the calculated frequency
distribution of fracture stresses as a function of the number of cracks N
1 J. I. Frenkel and T. A. Kontorova, /. Phijsics (U.S.S.R.), vol. 7, pp. 108114, 1943.
2 W. WeibuU, A Statistical Theory of the Strength of Materials, Roij. Swed. Inst.
Eng. Research, no. 151, 1939.
3 N. Davidenkov, E. Shevandin, and F. Wittman, J. Appl. Mech., vol. 14, pp
6367, 1946.
446
Applications to Materials Testing
[Chap. 16
on the basis of a Laplacian distribution of cracks. ^ Note that the larger
the total number of cracks the smaller the scatter in fracture stress
because of the higher probability of finding a critical defect. Also, as
the number of defects increases, the mean value of the fracture stress
decreases, but as the number of defects reaches large values, there is less
relative decrease in the mean value. Existing data are not numerous
enough to determine which of these distribution functions for crack size
is most generally applicable.
120
>.100
o
1 80
1 60
D
11)
^ 40
20

/I/
= 10'6


n/l/10^2



..

/
])
\J\
^^
^lO''
0.2 0.3
Relative fracture stress
0.4
0.5
Fig. 168. Calculated frequency distribution of fracture stress as a function of number
of cracks, A^. (J. C. Fisher and J. H. Hollomon, Trans. AIME, vol. 171, p. 555, 1947.)
Ductile metals do not exhibit so large a size effect as brittle materials.
The weakestlink concept is not well suited to this situation. The diffi
culty with using the weakestlink concept with ductile metals is that
plastic deformation alters both the size and the orientation of the defects,
and the individual volume elements can no longer be assumed to be
independent.
1613. Statistical Treatment of the Fatigue Limit
The statistical nature of the fatigue properties of metals has been
discussed in Chap. 12. In analyzing the results of a number of fatigue
specimens tested to failure at a constant stress, it is usual procedure to
assume that the logarithm of the number of cycles to failure is normally
distributed. The standard procedures for determining the mean and
standard deviation and the F test and t test for significant differences
can be used provided that all computations are based on log N instead of
A''. However, there are reasons to believe that the logarithmicnormal
' J. C. Fisher and J. H. Hollomon, Trans. AIME, vol. 171, pp. 546561, 1947.
Sec. 1613] Statistics Applied to Materials Testing 447
distribution does not accurately describe the distribution of fatigue life
at the upper and lower extremes of the distribution. Since fatigue life
depends on the weakest section rather than on average behavior, it is
to be expected that the extremevalue distribution function would pro
vide better agreement in these regions.
The statistical analysis of the fatigue data in the region of the fatigue
limit requires special techniques which have not yet been discussed in
this chapter. In making tests to measure the fatigue limit, we can test
a given specimen only at a particular stress, and if the specimen fails
before the cutoff limit of 10^ cycles then we know that the fatigue limit
of the specimen is somewhere below the stress level which we used.
Consequently, if the specimen does not fail (a runout) in the prescribed
number of cycles, we know that its fatigue limit lies somewhere above
the test stress. Because a given specimen cannot be tested more than
once, even if it "ran out," it is necessary to estimate the statistics of the
fatigue limit by testing a large number of presumably identical specimens
at different stress levels. Experiments of this type are known as sensi
tivity experiments. Two particular types of statistical analysis of sensi
tivity experiments have been applied to the statistical determination of
the fatigue limit. The procedures used will be briefly described below.
Stepbystep procedures for the analysis will be found in the ASTM
Manual on statistical analysis of fatigue data.^
The first of the methods for the statistical determination of the fatigue
limit is known as probit analysis. A number of specimens are tested at
each of four to six stress levels in the vicinity of the estimated fatigue
limit. When the percentage of the specimens which survived at each
stress is plotted on normalprobability paper against the stress level, a
straight line is obtained. One way of performing the experiment is to
test 20 specimens at each stress level. The analysis follows the pro
cedures given by Finney in his book on probit analysis.^ The percentage
of survivors are converted to probit values^ and a regression line is deter
mined according to Finney's procedure for weighting the values at differ
ent percentages of survival. However, the same type of analysis can be
made without resorting to any special techniques.
The number of specimens tested at each stress should be apportioned
with regard to the expected percentage of survival. Not fewer than
5 specimens should be tested at a stress level, and at least a total of
50 specimens should be used in the determination of the median value
1 "A Tentative Guide for Fatigue Testing and the Statistical Analysis of Fatigue
Data," ASTM Spec. Tech. Publ. 9lA, 1958.
2 D. J. Finney, "Probit Analysis," Cambridge University Press, New York, 1952.
^ The term probit is an abbreviation of probability unit. Transformation to probits
is a method of eliminating negative values from the standard normal deviate z.
448
Applications to Materials Testing
[Chap. 16
of fatigue limit. If 10 specimens are tested at stress levels which give
between 25 and 75 per cent survival, 15 specimens should be tested at
stresses which give 15 to 20 and 80 to 85 per cent survival and 20 speci
mens are required at stresses which result in 10 and 90 per cent survival.
With this relative distribution of specimens the test values are adequately
weighted at the extreme values, and it is possible to use a standard linear
regression analysis. In the standard linearregression equation X is the
stress level, and Y is the percentage of survivors converted to values of
the standard normal deviate z. The values of z corresponding to a given
percentage of survivors p are obtained by entering an expanded version
of Table 162 under the column headed Area with 1 — p/100. The
50
.48
a.
O
S46
44
42
40
X = failure
o = nonfailure
J I I I 1 I I I
J \ L
J I
2 4 6 8 10 12 14 16 18
Specimen number
Fig. 169. Staircase testing sequence for determination of mean fatigue limit.
median value of the fatigue limit (often loosely called the mean fatigue
limit) is given by the stress which intercepts the regression line at 50 per
cent survival. Its standard deviation is given by the stress range between
50 and 84 per cent survival.
The second method of analyzing fatigue data at the fatigue limit is the
staircase, or upanddown, method.^ This method provides a good meas
ure of the mean fatigue limit with fewer specimens than are required with
the probit method, but it has the disadvantage that tests must be run in
sequence. The testing procedure is illustrated in Fig. 169. The first
specimen is tested at the estimated value of the fatigue limit. If this
specimen fails, the stress for the next specimen is decreased by a fixed
increment. This procedure is continued for each succeeding specimen
until a runout is obtained. The stress applied to the next specimen is
then increased by the increment. The procedure is further continued,
1 W. J. Dixon and A. M. Mood, J. A^n. Statist. Assoc, vol. 43, p. 109, 1948.
1613]
Statistics Applied to Materials Testing
449
the stress being increased when a specimen runs out and decreased when
it fails. Fifteen to twentyfive specimens must be tested.
Table 169
Method of Analyzing Staircase Data
Stress,
psi
i
nonfailures
irii
I'^m
46,000
3
1
3
9
45,000
2
2
4
8
44,000
1
4
4
4
43,000
1
A' = 8
^ = 11
B = 21
d = stress increment = 1,000 psi
A'o = first stress level = 43,000 psi
X = 43,000 + 1,000(11.^ + }.i) = 44,870 psi
s = 1.620(1,000)
8 X 21  (11)2
+ 0.029
]■
240 psi
To determine the mean fatigue limit, the data are arranged in a tabular
form as in Table 169. A study of Fig. 169 shows that there are 10 fail
ures and 8 nonfailures out of the 18 specimens which were tested. Since
the analysis is based on the least frequent event (failures or nonfailures),
only the nonfailures are considered in Table 169. The lowest stress level
at which a nonfailure is obtained is denoted i = 0, the next i = 1, etc.
The mean fatigue limit X and its standard deviation s are determined
from Eqs. (1634) and (1635). The constants in these equations are
explained in Table 169. The positive sign is used in Eq. (1634) when
the analysis is based on nonfailures, while the negative sign is used when
it is based on failures.
X = Xo + d
s = 1.620rf
(NB_jA^
f 0.029
NB  A"
(1634)
> 0.3 (1635;
The staircase method provides a better estimate of the mean fatigue limit
than the probit method, and with a fewer number of specimens. How
ever, the latter method gives a better estimate of the standard devia
tion, and it provides overall a greater amount of information because the
probit curve is actually a response curve between percentage survival and
stress.
450 Applications to Materials Testing [Chap. 16
BIBLIOGRAPHY
Dixon, W. J., and F. J. Massey, Jr.: "Introduction to Statistical Analysis," 1st ed.,
McGrawHill Book Company, Inc., New York, 1951.
Fisher, R. S.: "Statistical Methods for Research Workers," 12th ed., Hafner Pub
lishing Company, New York, 1954.
Goulden, C. H. : "Methods of Statistical Analysis," 2d ed., John Wiley & Sons, Inc.,
New York, 1952.
Hald, A.: "Statistical Theory with Engineering Applications," John Wiley & Sons,
Inc., New York, 1952.
Olds, E. G., and Cyril Wells: Statistical Methods for Evaluating the Quality of Cer
tain Wrought Steel Products, Trans. ASM, vol. 42, pp. 845899, 1950.
Part Four
PLASTIC FORMING OF METALS
Chapter 17
GENERAL FUNDAMENTALS
OF METALWORKING
171. ClassiFication of Formins Processes
The importance of metals in modern technology is due, in large part,
to the ease with which they may be formed into useful shapes. Hundreds
of processes have been developed for specific metalworking applications.
However, these processes may be classified into only a few categories on
the basis of the type of forces applied to the work piece as it is formed
into shape. These categories are:
1. Directcompressiontype processes
2. Indirectcompression processes
3. Tensiontype processes
4. Bending processes
5. Shearing processes
In directcompression processes the force is applied to the surface
of the workpiece, and the metal flows at right angles to the direction
of the compression. The chief examples of this type of process are
forging and rolling (Fig. 171). Indirectcompression processes include
wire and tube drawing, extrusion, and the deep drawing of a cup.
The primary applied forces are frequently tensile, but the indirect com
pressive forces developed by the reaction of the workpiece with the die
reach high values. Therefore, the metal flows under the action of a com
bined stress state which includes high compressive forces in at least one
of the principal directions. The best example of a tensiontype forming
process is stretch forming, where a metal sheet is wrapped to the contour
of a die under the application of tensile forces. Bending involves the
application of bending moments to the sheet, while shearing involves the
application of shearing forces of sufficient magnitude to rupture the metal
in the plane of shear. Figure 171 illustrates these processes in a very
simplified way.
453
454
Plastic Forming of Metals
[Chap. 17
Plastic forming operations are performed for at least two reasons. One
objective is to produce a desired shape. The second objective is to
improve the properties of the material through the alteration of the
distribution of microconstituents, the refinement of grain size, and the
introduction of strain hardening.
Forging
Rolling
Wire drawing
Extrusion
t t A
Deep drawing
^
^■\
Stretch forming Bending
Fig. 1 71 . Typical forming operations.
Shearing
Plastic working processes which are designed to reduce an ingot or
billet to a standard mill product of simple shape, such as sheet, plate,
and bar, are called primary mechanical working processes. Forming meth
ods which produce a part to a final finished shape are called secondary
mechanical working processes. Most sheetmetal forming operations, wire
drawing, and tube drawing are secondary processes. The terminology in
this area is not very precise. Frequently the first category is referred to
as processing operations, and the second is called fabrication.
An important purpose of plastic working operations is to break down
and refine the columnar or dendritic structure present in cast metals and
Sec. 1 72] General Fundamentals of Metalworking 455
alloys. Frequently the low strength and ductility of castings are due to
the presence of a brittle constituent at the grain boundaries and dendritic
boundaries. By compressive deformation it is often possible to fragment
a brittle microconstituent in such a way that the ductile matrix flows
into the spaces between fragments and welds together to leave a per
fectly sound structure. Once the brittle constituent is broken up, its
effect on the mechanical properties is minor and ductility and strength
are increased. Forging and rolling are the processes ordinarily used for
breaking down a cast structure. However, extrusion is the best method
because the billet is subjected to compressive forces only.
172. Effect of Temperature on Forming Processes
Forming processes are commonly classified into hotworking and cold
working operations. Hot working is defined as deformation under cou'
ditions of temperature and strain rate such that recovery processes take
place simultaneously with the deformation. On the other hand, cold
working is deformation carried out under conditions where recovery proc
esses are not effective. In hot working the strain hardening and distorted
grain structure produced by deformation are very rapidly eliminated by
the formation of new strainfree grains as the result of recrystallization.
Very large deformations are possible in hot working because the recovery
processes keep pace with the deformation. Hot working occurs at an
essentially constant flow stress, and because the flow stress decreases with
increasing temperature, the energy required for deformation is generally
much less for hot working than for cold working. Since strain hardening
is not relieved in cold working, the flow stress increases with deformation.
Therefore, the total deformation that is possible without causing fracture
is less for cold working than for hot working, unless the effects of cold
work are relieved by annealing.
It is important to realize that the distinction between cold working and
hot working does not depend upon any arbitrary temperature of defor
mation. For most commercial alloys a hotworking operation must be
carried out at a relatively high temperature in order that a rapid rate of
recrystallization be obtained. However, lead and tin recrystallize rapidly
at room temperature after large deformations, so that the working of these
metals at room temperature constitutes hot working. Similarly, working
tungsten at 2,000°F, in the hotworking range for steel, constitutes cold
working because this highmelting metal has a recrystallization temper
ature above this working temperature.
Hot Working
Hot working is the initial step in the mechanical working of most
metals and alloys. Not only does hot working result in a decrease in
456 Plastic Forming of Metals [Chap. 17
the energy required to deform the metal and an increased abihty to flow
without cracking, but the rapid diffusion at hotworking temperatures
aids in decreasing the chemical inhomogeneities of the castingot struc
ture. Blowholes and porosity are eliminated by the welding together of
these cavities, and the coarse columnar grains of the casting are broken
down and refined into smaller equiaxed recrystallized grains. These
changes in structure from hot working result in an increase in ductility
and toughness over the cast state.
However, there are certain disadvantages to hot working. Because
high temperatures are usually involved, surface reactions between the
metal and the furnace atmosphere become a problem. Ordinarily hot
working is done in air, oxidation results, and a considerable amount of
metal may thus be lost. Reactive metals like molybdenum are severely
embrittled by oxygen, and therefore they must be hotworked in an inert
atmosphere or protected from the air by a suitable container. Surface
decarburization of hotworked steel can be a serious problem, and fre
quently extensive surface finishing is required to remove the decarburized
layer. Rolledin oxide makes it difficult to produce good surface finishes
on hotrolled products, and because allowance must be made for expan
sion and contraction, the dimensional tolerances for hotworked mill
products are greater than for coldworked products. Further, the struc
ture and properties of hotworked metals are generally not so uniform
over the cross section as in metals which have been coldworked and
annealed. Since the deformation is always greater in the surface layers,
the metal will have a finer recrystallized grain size in this region. Because
the interior will be at higher temperatures for longer times during cooling
than will be the external surfaces, grain growth can occur in the interior
of large pieces, which cool slowly from the working temperature.
The lower temperature limit for the hot working of a metal is the
lowest temperature at which the rate of recrystallization is rapid enough
to eliminate strain hardening in the time when the metal is at temper
ature. For a given metal or alloy the lower hotworking temperature
will depend upon such factors as the amount of deformation and the time
when the metal is at temperature. Since the greater the amount of defor
mation the lower the recrystallization temperature, the lower temperature
limit for hot working is decreased for large deformations. Metal which
is rapidly deformed and cooled rapidly from temperature will require a
higher hotworking temperature for the same degree of deformation than
will metal slowly deformed and slowly cooled.
The upper limit for hot working is determined by the temperature at
which either melting or excessive oxidation occurs. Generally the maxi
mum working temperature is limited to 100°F below the melting point.
This is to allow for the possibility of segregated regions of lowermelting
172]
General Fundamentals of Metalworking 457
point material. Only a very small amount of a grainboundary film of
a lowermelting constituent is needed to make a material crumble into
pieces when it is deformed. Such a condition is known as hot shortness,
or burning.
Most hotworking operations are carried out in a number of multiple
passes, or steps. Generally the working temperature for the intermediate
passes is kept well above the minimum working temperature in order to
take advantage of the economies offered by the lower flow stress. It is
likely that some grain growth will occur subsequent to the recrystalli
zation at these temperatures. Since a finegrainsized product is usually
desired, common practice is to lower the working temperature for the
last pass to the point where grain growth during cooling from the work
ing temperature will be negligible. This finishing temperature is usually
just above the minimum recrystallization temperature. In order to
ensure a fine recrystallized grain size, the amount of deformation in the
last pass should be relatively large.
Cold Working
As was shown in Sec. 59, cold working of a metal results in an increase
in strength or hardness and a decrease in ductility. When cold working
is excessive, the metal will fracture before reaching the desired size and
shape. Therefore, in order to avoid such difficulties, coldworking oper
ations are usually carried out in several steps, with intermediate anneal
ing operations introduced to soften the coldworked metal and restore the
ductility. This sequence of repeated cold working and annealing is fre
quently called the coldworkanneal cycle. Figure 172 schematically
illustrates the property changes involved in this cycle.
Although the need for annealing operations increases the cost of form
ing by cold working, particularly for reactive metals which must be
annealed in vacuum or inert atmospheres, it provides a degree of versa
tility which is not possible in hotworking operations. By suitably
Cold work
Annealing
Per cent cold work
Annealing temperature
Fig. 172. Typical variation of strength and ductility in the coldworkanneal cycle.
458 Plastic Forming of Metals [Chap. 17
adjusting the eoldwork anneal cycle the part can be produced with any
desired degree of strain hardening. If the finished product must be
stronger than the fully annealed material, then the final operation must
be a coldworking step with the proper degree of deformation to produce
the desired strength. This would probably be followed by a stress relief
to remove residual stresses. Such a procedure to develop a certain
combination of strength and ductility in the final product is more suc
cessful than trying to achieve the same combinations of properties by
partially softening a fully coldworked material, because the recrystalli
zation process proceeds relatively rapidly and is quite sensitive to small
temperature fluctuations in the furnace. If it is desired to have the final
part in the fully softened condition, then an anneal follows the last cold
working step.
It is customary to produce coldworked products like strip and wire in
different tempers, depending upon the degree of cold reduction following
the last anneal. The coldworked condition is described as the annealed
(soft) temper, quarterhard, halfhard, threequarterhard, fullhard, and
spring temper. Each temper condition indicates a different percentage
of cold reduction following the annealing treatment.
1 73. Effect of Speed of Deformation on Forming Processes
The response of a metal to forming operations can be influenced by the
ispeed with which it is deformed. The existence of a transition from a
ductile to brittle condition for most bodycentered cubic metals over a
certain temperature range has already been discussed in earlier chapters.
This transitiontemperature phenomenon is more pronounced for rapid
rates of deformation. Thus, for certain metals there may be a temper
ature region below which the metal will shatter when subjected to a high
speed or impact load. For example, iron and steel will crack if hammered
at temperatures well below room temperature, whereas a limited amount
of slowspeed deformation can be accomplished in the same temperature
range.
Table 171 lists typical values of velocities for different types of testing
and forming operations. It is important to note that the forming veloc
ity of most commercial equipment is appreciably faster than the cross
head velocity used in the standard tension test. Therefore, values of
flow stress measured in the tension test are not directly applicable for
the computation of forming loads. For cold working a change in the
forming speed of several orders of magnitude results in only about a
20 per cent increase in the flow curve, so that for practical purposes the
speed of deformation can be considered to have little effect. Exceptions
to this statement are the possibility of brittle behavior in a certain tem
p<jrature region at high forming speeds and the fact that highspeed
Sec. 1 74] General Fundamentals of Metalworking 459
deformation accentuates the yieldpoint phenomenon in mild steel.
Highspeed deformation may produce regions of nonuniform deformation
(stretcher strains) in a steel sheet which shows no stretcher strains on
slowspeed deformation.
Table 171
Typical Values of Velocity Encountered in
Different Testing and Forming Operations
Operation Velocity, ft /sec
Tension test 2 X lO"" to 2 X IQ^
Hydraulic extrusion press 0.01 to 10
Mechanical press . 5 to 5
Charpy impact test 10 to 20
Forging hammer 10 to 30
Explosive forming 100 to 400
The flow stress for hot working is quite markedly affected by the speed
of deformation. There are no routine methods for measuring the flow
stress during hot working. However, a highspeed compression testing
machine, called the cam plastometer, is one device which has given good
results. Data obtained^ for copper, aluminum, and mild steel at a series
of constant temperatures indicate that the dependence of flow stress on
strain rate over a range of e from 1 to 40 sec~' can be expressed by Eq.
(935). Because the time at temperature is shorter for high forming
speeds, the minimum recrystallization temperature is raised and the
minimum hotworking temperature will be higher. On the other hand,
because the metal retains the heat developed by deformation to a greater
extent at high forming speeds, there is a greater danger of hot shortness.
For metals which have a rather narrow hotworking range these opposing
effects serve to close the gap still further and make it impractical to
carry out hot working at very high forming speeds.
A recent development has been the ultrahighvelocity forming of metals
at velocities of 100 to 400 ft /sec by using the energy generated by the
detonation of explosives.^ Among the advantages offered are the fact
that highstrength materials can be formed without large springback,
that metals flow readily into the recesses of the die, and that certain
shapes can be produced which cannot be made by any other technique.
1 74. Effect of Metallurgical Structure on Formins Processes
The forces required to carry out a forming operation are directly related
to the flow stress of the metal being worked. This, in turn, depends on
1 J. F. Alder and V. A. Phillips, /. Inst. Metals, vol. 81, pp. 8086, 195455; J. E.
Hockett, Proc. ASTM, vol. 59, pp. 13091319, 1959.
2 T. C. DuMond, Metal Prog., vol. 74, pp. 6876, 1958.
460 Plastic Formins of Metals [Chap. 17
the metallurgical structure and composition of the alloy. For pure
metals the ease of mechanical working will, in general, decrease with
increasing melting point of the metal. Since the minimum recrystalli
zation temperature is approximately proportional to the melting point,
the lower temperature limit for hot work will also increase with melting
point. (As a rough approximation, this temperature is about onehalf
the melting point.) The addition of alloying elements to form a solid
solution alloy generally raises the flow curve, and the forming loads
increase proportionately. Since the melting point is often decreased by
solidsolution alloying additions, the upper hotworking temperature
must usually be reduced in order to prevent hot shortness.
The plastic working characteristics of twophase (heterogeneous) alloys
depend on the microscopic distribution of the secondphase particles.
The presence of a large volume fraction of hard uniformly dispersed
particles, such as are found in the SAP class of hightemperature alloys,
greatly increases the flow stress and makes working quite difficult. If
the secondphase particles are soft, they have only a small effect on the
working characteristics. If these particles have a lower melting point
than the matrix, then difficulties with hot shortness will be encountered.
The presence of a massive, uniformly distributed microconstituent, such
as pearlite in mild steel, results in less increase in flow stress than for
very finely divided secondphase particles. The shape of the carbide
particles can be important in coldworking processes. For annealed steel,
a spheroidization heat treatment, which converts the cementite platelets
to spheroidal cementite particles, is often used to increase the formability
at room temperature. An important exception to the general rule that
the presence of a hard second phase increases the difficulty of forming is
brass alloys containing 35 to 45 per cent zinc (Muntz metal). These
alloys, which consist of a relatively hard beta phase in an alphabrass
matrix, actually have lower flow stresses in the hotworking region than
the singlephase alphabrass alloys. In the coldworking region the flow
stress of alphabeta brass is appreciably higher than that of alpha brass.
Alloys which contain a hard second phase located primarily at the grain
boundaries present considerable forming problems because of the tend
ency for fracture to occur along the grain boundaries.
As the result of a mechanical working operation secondphase particles
will tend to assume a shape and distribution which roughly correspond
to the deformation of the body as a whole. Secondphase particles or
inclusions which are originally spheroidal will be distorted in the principal
working direction into an ellipsoidal shape if they are softer and more
ductile than the matrix. If the inclusions or particles are brittle, they
will be broken into fragments which will be oriented parallel to the work
ing direction, while if the particles are harder and stronger than the
Sec. 174] General Fundamentals of Metalworking 461
matrix, they will be essentially undeformed.' The orientation of second
phase particles during hot or cold working and the preferred fragmen
tation of the grains during cold working are responsible for the fibrous
structure which is typical of wrought products. The fiber structure can
be observed after macroetching (Fig. 173). Microscopic examination
of wrought products frequently shows the results of this mechanical fiber
ing (Fig. 174). An important consequence of mechanical fibering is that
the mechanical properties may be different for different orientations of
the test specimen with respect to the fiber (working) direction. In
Fig. 173. Fiber structure in wrought product revealed by macroetching. 5X
general, the tensile ductility, fatigue properties, and impact properties
will be lower in the transverse direction (normal to the fiber) than in the
longitudinal direction.
The forming characteristics of an alloy can be affected if it undergoes
a straininduced precipitation or straininduced phase transformation.
If a precipitation reaction occurs in a metal while it is being formed, it will
produce an increase in the flow stress but, more important, there will be
an appreciable decrease in ductility, which can result in cracking. When
brittleness is caused by precipitation, it usually results when the working
is carried out at a temperature just below the solvus line or from cold
1 H. Unkel, J. Inst. Metals, vol. 61, pp. 1711<)G, 1937; F. B. Pickering, J. Iron
Steel Inst. (London), vol. 189, pp. 148159, 1958.
2 The boundary on the phase diagram of the limit of solid solubility of a solid solu
tion.
462 Plastic Forming of Metals [Chap. 17
working after the alloy had been heated to the same temperature region.
Since precipitation is a diffusioncontrolled process, difficulty from this
factor is more likely when forming is carried out at a slow speed at an
elevated temperature. To facilitate the forming of agehardenable
aluminum alloys, they are frequently refrigerated just before forming
in order to suppress the precipitation reaction.
A most outstanding practical example of a straininduced phase trans
formation occurs in certain austenitic stainless steels where the Cr:Ni
ratio results in an unstable austenite phase. When these alloys are cold
worked, the austenite transforms to ferrite along the slip lines and pro
«* r>*
Fi3. 1 74. Fibered and banded structure in logitudinal direction of a hotrolled mild
steel plate. 100 X.
duces an abnormal increase in the flow stress for the amount of defor
mation received. While this phase transformation is often used to
increase the yield strength by cold rolling, it can also result in cracking
during forming if the transformation occurs in an extreme amount in
regions of highly localized strain.
1 75. Mechanics of Metal Forming
One of the prime objectives of research in metal forming is to express
the forces and deformations involved in forming processes in the mathe
matical language of applied mechanics so that predictions can be made
about the forces required to produce the desired shape. Since the forces
Sec. 175] General Fundamentals of Metalworking 463
and deformations are generally quite complex, it is usually necessary to
use simplifying assumptions to obtain a tractable solution. This branch
of mechanics is an outgrowth of the theory of plasticity discussed in
Chap. 3, which should be reviewed before completing this chapter and
the remaining chapters of this book.
Plasticity theory has made important advances since World War II,
and numerous efforts have been made to utilize the new knowledge to
improve the accuracy of prediction of the metalforming equations.
Most of this work involves the use of the slipfield theory discussed briefly
in Sec. 312. While in many instances the use of these more advanced
methods provides a deeper insight into the forming process, usually this
additional information cannot be obtained without considerable increase
in mathematical complexity. Because an analysis based on slipfield
theory cannot be made with the background provided in Chap. 3, in dis
cussing the mathematical aspects of specific forming processes in succeed
ing chapters only the simpler equations will be derived. References to
more advanced treatments will be given wherever this is applicable.
The mathematical descriptions of technologically important forming
processes which were developed by SiebeP and other German workers
were based on the assumption that the maximumshearstress law was
the proper criterion for describing the stress condition for producing
plastic flow.
0"!
(171)
Subsequent work showed that the distortionenergy, or Von Mises, flow
criterion [Eq. (172)] provided better agreement with experimental data.
(cTi  cr.>)2 + (a,  as)' + (era  cti)^ = 2^0^ (172)
Therefore, the distortionenergy criterion for yielding is to be preferred
and will be used in most of the analyses of forming processes presented in
subsequent chapters. However, as was seen in Sec. 34, the distortion
energy and maximumshearstress criteria differ at most by only 15 per
cent, and in view of the uncertainties present in the analysis of some of
the complex forming operations the tw^o yield criteria can be considered
nearly equivalent. Therefore, the maximumshearstress law will be
used in certain cases where it provides appreciable simplification to the
analysis.
An important feature of plasticity theory is the assumption that the
1 An excellent review of E. Siebel's work is available in an English translation by
J. H. Hitchcock; it appeared weekly in the magazine Steel, from Oct. 16, 1933, to
May 7, 1934.
2 In much of the literature in this field the flow stress oq is denoted by the symbol k.
In this text k is taken to indicate the ^I'eld stress in shear.
464 Plastic Forming of Metals [Chap. 17
introduction or removal of a hj^drostatic or mean stress has no effect on
the flow stress or the state of strain. It is considered that only the stress
deviator (see Sec. 214) is of importance in producing plastic flow, and it
is this stress term which appears in the plasticity equations [see Eq.
(342)]. This assumption is borne out by the experimental fact that the
yield stress for the beginning of flow is independent of the mean stress.
However, at large plastic strains the hydrostatic stress does have an effect
on the flow stress. For increasing hydrostatic stress, the flow curve at
large strains is raised. Moreover, as was shown in Sec. 716, the ductility
of metals in tension is appreciably increased when a high hydrostatic
pressure is present. This explains why nominally brittle materials may
often be extruded successfully, since a high hydrostatic compression is
developed owing to the reaction of the workpiece with the extrusion
container.
Because large deformations occur in metal forming, it is important to
express stress and strain as true stress and true, or natural, strain. To a
very good approximation it is permissible to assume that the volume
remains constant during deformation. This leads to the convenient
relationships
€1 + €2 + €3 = ('[7 ^^
or dei + de2 + des =
Frequently it can be assumed that the strain increment is proportional
to the total strain. This is called proportional straining and leads to the
following equation, which is often useful for integrating equations con
taining the strain differential.
dei ^ de2 ^ des (\7 A)
ei 62 63
A basic premise of plasticity theory is that equivalent strain hardening
is obtained for an equivalent tensile or compressive deformation. For a
tensile strain ei equal to a compressive strain €3, we can write
ei = —63 = In^ = ~lnr = In t
ho
hi __
ho
= 1 +
hi — ho
ho
From
constancy of volume
Lo
Ai
Li
Ao
Therefore
hq
 h,
7_
_ Ao  Ai
Ao
Ao  Ai
(175)
Sec. 175]
General Fundamentals of Metalworking 465
Equation (175) expresses the fact that for equal true strains the reduc
tion of area is equal to the reduction in height or thickness. It is fre
quently useful to employ these parameters as a substitute for strain in
metalforming experiments.
The flow curve (true stressstrain curve) determined for either tension
or compression is the basic relationship for the strainhardening behavior
of the material. It is used to determine the value of the flow stress oq
for calculating forming loads. The value of the flow stress will of course
depend on the temperature, the speed of deformation, and possibly the
existence of a straininduced transformation, as described in earlier sec
tions of this chapter. For most commercial forming operations the
10 20 30 40 50
Reduction of area by drawing, %
True strain €
Fig. 175. Flow curve constructed from Fig. 176. Method of using average flow
stressstrain curves after different amounts stress to compensate for strain hardening,
of reduction.
degree of strain hardening which occurs for a given reduction is higher
than would be determined from a tensileflow curve. This is due to the
fact that the metal undergoes nonuniform flow during deformation
because it is not allowed to flow freely. The lightly deformed regions
provide a constraint to plastic flow, just as in the case of a notch in a
tension specimen (Sec. 712), and the flow stress is raised. One way of
determining the flow curve in cases where deformation is nonvniiform is to
determine the yield stress after different amounts of reduction in the
forming operation (Fig. 175). A method for measuring the flow stress
for coldworked metals, which is used frequently in England,^ is to meas
ure the pressure required to produce plastic flow when a sheet is com
pressed between two rigid anvils. In this test the metal is subjected to
plane compression, since there is no deformation in the width direction.
1 A. B. Watts and H. Ford, Proc. Inst. Mech. Engrs. {London), vol. IB, pp. 448453,
195253.
466 Plastic Forming of Metals [Chap. 17
Flow curves for a number of steels and nonferrous metals have been
obtained^ by this method.
In many plastic forming operations, such as extrusion, the strains are
much greater than can be obtained in a tension or compression test.
It is possible to get good estimates of the flow stress for reductions greater
than 70 to 80 per cent by linear extrapolation when ctq is plotted against
the logarithm of the strain or the reduction in area.^
For hot working the metal approaches an ideal plastic material, and
the flow stress is constant and independent of the amount of deformation
at a given temperature and speed of deformation. To allow for strain
hardening in cold working, it is customary to use a constant value of
flow stress which is an average over the total strain, as in Fig. 176. An
alternative, which adds to the mathematical complexity, is to include a
mathematical expression for the flow curve in the analysis. Usually this
is limited to a simple power function like Eq. (31).
To describe the plastic flow of a metal, it is just as important to
describe the geometry of flow in relation to the stress system as it is to
be able to predict the stress conditions to produce plastic flow. A basic
assumption of plasticity theory which allows this is that at any stage in
the deformation process the geometry of strain rates is the same as the
geometry of stress, i.e., that stress and strain are coaxial. This is a good
assumption up to moderate strains, but at large strains, where preferred
orientations may have been developed, the stress and strain systems usu
ally are not identical. If Lode's stress and strain parameters are equal
(see Sec. 35), then the plastic stress and strain can be considered coaxial.
Since metalforming problems are concerned with large strains of the
order of unity, elastic strains of the order of 0.001 are negligible by com
parison and the metal can be treated as a rigid plastic material. Regions
of the metal which have been strained only elastically and regions between
the elavsticplastic boundary in which the yield stress has been exceeded
but flow is constrained by the elastic region are considered to be rigid.
Only flow in the completely plastic region of the body is considered in
the relatively simple analyses of plastic forming given in succeeding
chapters. By using the more advanced slipfield theory, it is possible
to consider the stress and strain in both the elastic and plastic regions,
and also along the elasticplastic boundary.
1 76. Work of Plastic Deformation
The total work required to produce a shape by plastic deformation
can be broken down into a number of components. The work of defor
1 R. B. Sims, J. Iron Steel Inst. {London), vol. 177, pp. 393399, 1954.
2 R. J. Wilcox and P. W. Whitton, J. Inst. Metals, vol. 88, pp. 145149, 19591960.
Sec. 1 76] General Fundamentals of Metalworking 467
mation Wd is the work required for homogeneous reduction of the voUime
from the initial to final cross section by uniform deformation. Often
part of the total work is expended in redundant work Wr. The redundant,
or internaldeformation, work is the energy expended in deforming the
body which is not involved in a pure change in shape. Finally, part of
the total work must be used to overcome the frictional resistances at the
interface between the deforming metal and the tools. Therefore, the
total work can be written as the summation of three components.
Wt = Wd + Wr + W, (176)
From the above definitions, it can be seen that the work of deformation
represents the minimum energy which must be expended to carry out a
particular forming process. This is equal to the area under the effective
stressstrain curve multiplied by the total volume.
Wd = Vja dl (177)
The efficiency of a forming process is the work of deformation divided
by the total work of deformation.
W^
Efficiency = r, = ^ (178)
W
The total work is usually measured with a wattmeter attached to the
electrical drive of the forming equipment. Typical efficiencies for form
ing processes 1 are extrusion 30 to 60 per cent, wire drawing 50 to 75 per
cent, and sheet rolling 75 to 95 per cent.
The following analysis' for the work of plastic deformation assumes
that the metal is an ideal plastic material which obeys the distortion
energy criterion of yielding and the LevyVon Mises theory of plastic
flow (Sec. 39). dWd is the increment of work per unit volume dissipated
during the infinitesimal straining of increments dei, dc2, and des.
dWd = (Xl dei + (To de, + (T3 d€3 (179)
Since constancy of volume exists, de^ = —dei — de2.
dWd = ((Ti  as) rfei + {ai  <Ji) de2 (1710)
From Eq. (348) it follows that
de, = deJ ;""'  ""'  "' ) (1711)
1 J. G. Wistreich, J. Iron Steel Insl. (Lundon), vol. 180, p. 54, 1955.
2 O. Hoffman and G. Sachs, "Introduction to the Theory of Plasticity for Engi
neers," pp. 5254, McGrawHill Book Company, Inc., New York, 1953.
468 Plastic Forming of Metals [Chap. 17
Introducing Eq. (1711) into Eq. (1710) and simplifying,
2(7i — 'T2 — (Xz
Upon referring to Eqs. (312) and (347) it is possible to simplify the
above equation to
dWa = ^dt  :4713)
oA
To integrate this equation, the variable X must be known as a function
of strain. X is given in terms of effective stress and strain by
_ a dt
3 de
and since the effective stress a is equal to the uniaxial yield stress in the
distortionenergy yield criterion, ao, we can write
X = ^° (1714)
o ae
Substituting into Eq. (1713),
dWa = ^^^ = aode (1715)
o Zero dt
From the definition of effective strain
a/2
rfe = V [{dei  de^y + ((ies  de,)^ + (^€3  de^)']^^ (1716)
o
and from the constancyofvolume relationship
des = —dei — de2
Simplifying Eq. (1716),
2
de = —7 (dei^ + de, de^ + de^'^f'^ (1717)
For the case of proportional straining, Eq. (1715) can be integrated to
obtain the total ideal work per unit volume.
Wa= f dWa=^ ^ Toie,' + e^e, + e^)^^ (1718)
1 77. Formability Tests and Criteria
The success of a forming operation depends on the application of high
enough forces to overcome the resistance to deformation without exhaust
Sec. 177] General Fundamentals of Metalworking 469
ing the ductility of the metal. It is only natural that attempts have been
made to predict the success of forming operations on the basis of simple
laboratory tests. Reliable information on the flow stress required to pro
duce deformation can be obtained with the tension or compression test
for slow rates of deformation, but there is no convenient test for exactly
determining these data at high rates of deformation.
Because of the difficulty of obtaining reliable data on the deformation
resistance of metals during hot working at high deformation rates a num
ber of empirical tests have been developed to evaluate the hot formability
of materials. The singleblow impact test^ has been used for many years
to evaluate whether or not a material can be hotworked without crack
ing. Good correlation with hotworking operations, such as forging and
piercing of solid rounds to make seamless tubes, has b